N[(1/2)-k*(k-1/2)*(-3)/Factorial[k], 10000]

Martin M. Musatov, 11/14/08, mmm@xxxxxx
N[(1/2)-k*(k-1/2)*(-3)/Factorial[k], 10000]
0.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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00+1/
k!
3.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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0
(-0.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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0000+k)
k
..5+1/k(3)(-.5+k)k
0.5`+3 (-0.5`+k)[{l.5, k, 2, 3}]
n a(n)
1 1
2 6
3 21
4 107

[1,6,21,107]

1.120529165250997236305196466566 n^2
1
12
63
428
%I A000045 M0692 N0256
%S A000045
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,%T
A000045
10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,%U
A000045
1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169
%N A000045 Fibonacci numbers:F(n)=F(n-1)+F(n-2),F(0)=0,F(1)=1,F(2)
=1,...
%C A000045 Also called Lam{\'e}'s sequence.%C A000045 F(n+2)=number of
binary sequences of length n that have no consecutive 0's.%C A000045 F
(n+2)=number of subsets of {1,2,...,n} that contain no consecutive
integers.%C A000045 F(n+1)=number of tilings of a 2 X n rectangle by 2
X 1 dominoes.%C A000045 F(n+1)=number of matchings in a path graph on
n vertices:F(5)=5 because the matchings
of the path graph on the vertices A,B,C,D are the empty set,{AB},{BC},
{CD},and {AB,CD}.-Emeric Deutsch (deutsch(AT)duke.poly.edu),Jun 18
2001
%C A000045 F(n)=number of compositions of n+1 with no part equal to 1
[Grimaldi]
%C A000045 Positive terms are the solutions to z=2xy^4+(x^2)y^3-2(x^3)
y^2-y^5-(x^4)y
+2y for x,y>=0 (Ribenboim,page 193).When x=F(n),y=F(n+1)
and z>0 then z=F(n+1).%C A000045 For Fibonacci search see Knuth,Vol.
3;Horowitz and Sahni;etc.%C A000045 F(n) is the diagonal sum of the
entries in Pascal's triangle at 45 degrees slope.-Amarnath Murthy
(amarnath_murthy(AT)yahoo.com),Dec 29 2001
%C A000045 F(n+1) is the number of perfect matchings in ladder graph
L_n=P_2 X P_n,-Sharon
Sela (sharonsela(AT)hotmail.com),May 19 2002
%C A000045 F(n+1)=number of (3412,132)-,(3412,213)-,and (3412,321)-
avoiding involutions
in S_n.%C A000045 This is also the Horadam sequence (0,1,1,1).-Ross La
Haye (rlahaye(AT)new.rr.com),Aug 18 2003
%C A000045 An INVERT transform of A019590.INVERT([1,1,2,3,5,8,...])
gives A000129.INVERT([1,2,3,5,8,13,21,...]) gives A028859.-Antti
Karttunen,Dec 12,2003
%C A000045 Number of meaningful differential operations of the k-th
order on the space R^3.
-Branko Malesevic (malesevic(AT)kiklop.etf.bg.ac.yu),Mar 02 2004
%C A000045 F(n)=number of compositions of n-1 with no part greater
than 2. Example:F(4)=3
because we have 3=1+1+1=1+2=2+1.
%C A000045 F(n)=number of compositions of n into odd parts;e.g.F(6)
counts 1+1+1+1+1+1,1+1+1+3,1+1+3+1,1+3+1+1,1+5,3+1+1+1,3+3,5+1.-Clark
Kimberling
(ck6(AT)evansville.edu),Jun 22 2004
%C A000045 F(n)=number of binary words of length n beginning with 0
and having all runlengths
odd;e.g.F(6) counts
010101,010111,010001,011101,011111,000101,000111,000001.-Clark
Kimberling (ck6(AT)evansville.edu),Jun 22
2004
%C A000045 F(n)=number of Catalan paths between the lines y=0 and y=3
from (0,0) to (n,GCD(n,2)).-Clark Kimberling (ck6(AT)
evansville.edu),Jun 22 2004
%C A000045 F(n)=number of (s(0),s(1),...s(n)) such that 0<s(i)<5, |s
(i)-s(i-1)|=1,and s(0)=1;
e.g.F(6) counts
121212,121232,121234,123212,123232,123234,123432,1223434.-Clark
Kimberling (ck6(AT)evansville.edu),Jun 22 2004
%C A000045 A relationship between F(n) and the Mandelbrot set is
discussed in the link'Le
nombre d'or dans l'ensemble de Mandelbrot' (in French).-Gerald
McGarvey (Gerald.McGarvey(AT)comcast.net),Sep 19 2004
%C A000045 For n>0,the continued fraction for F(2n-1)*Phi=[F(2n);L
(2n-1),L(2n-1),L(2n-1),...] and the continued fraction for F(2n)*Phi=[F
(2n+1);L(2n)-2,L(2n)-2,L(2n)-2,...] where L(i) is the i-th Lucas
number (A000204).-Clark Kimberling (ck6(AT)evansville.edu),Nov 28 2004
%C A000045 F(n)=number of permutations p of 1,2,3,...,n such that|k-p
(k)|<=1 for k=1,2,...,n.(For<=2 and<=3,see A002524 and A002526.).-
Clark Kimberling
(ck6(AT)evansville.edu),Nov 28 2004
%C A000045 The ratios F(n+1)/F(n) for n>0 are the convergents to the
simple continued fraction
expansion of the golden section.-Jonathan Sondow (jsondow(AT)
alumni.princeton.edu),Dec 19 2004
%C A000045 Lengths of successive words (starting with a) under the
substitution:{a->ab,b->a}-J.F.J.Laros (jlaros(AT)liacs.nl),Jan 22 2005
%C A000045 The Fibonacci sequence,like any additive sequence,naturally
tends to be geometric
with common ratio not a rational power of 10;consequently,for a
sufficiently large number of terms,Benford's law of first significant
digit {i.e.,first digit 1=<d=<9 occurring with probability log_10(d+1)-
log_10(d)} holds.-Lekraj Beedassy (blekraj(AT)yahoo.com),Apr
29 2005
%C A000045 a(n)=Sum(abs(A108299(n,k)):0<=k<=n).-Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com),Jun 01 2005
%C A000045 a(n)=A001222(A000304(n)).%C A000045 Fib(n+2)=sum
(k=0..n,binomial(floor((n+k)/2),k)),row sums of A04685 4.-Paul
Barry (pbarry(AT)wit.ie),Mar 11 2003
%C A000045 Number of order ideals of the "zig-zag" poset.See vol.1,ch.
3,prob.23 of Stanley.-Mitch Harris (Harris.Mitchell (AT)
mgh.harvard.edu),Dec 27,2005
%C A000045 F(n+1)/F(n) is also the Farey fraction sequence (see
A097545 for explanation) for
the golden ratio,which is the only number whose Farey fractions
and continued fractions are the same.-Joshua Zucker (joshua.zucker(AT)
stanfordalumni.org),May 08 2006
%C A000045 a(n+2) is the number of paths through 2 plates of glass
with n reflections (reflections occurring at plate/plate or plate/air
interfaces).Cf.A006356-A006359.-Mitch Harris (Harris.Mitchell(AT)
mgh.harvard.edu),Jul 06 2006
%C A000045 F(n+1) equals the number of downsets (i.e.decreasing
subsets)of an n-element fence,i.e.an ordered set of height 1 on
{1,2,...,n} with 1>2<3>4
<... n and no other comparabilities.Alternatively,F(n
+1) equals
the number of subsets A of {1,2,...,n} with the property that,if
k is in A,then the adjacent elements of {1,2,...,n} belong to
A,i.e.both k-1 and k+1 are in A (provided they are in {1,2,...,n}).-
Brian A.Davey (B.Davey(AT)latrobe.edu.au),Aug 25 2006
%C A000045 Number of Kekule structures in polyphenanthrenes.See the
paper by Lukovits and
Janezic for details.-Parthasarathy Nambi (PachaNambi(AT)yahoo.com),Aug
22 2006
%C A000045 Inverse:With phi=(sqrt(5)+1)/2,round(log_phi(sqrt((sqrt(5) a
(n)+sqrt(5 a(n)^2-4))(sqrt(5) a(n)+sqrt(5 a(n)^2+4)))/2))=n for
n>=3,obtained
by rounding the arithmetic mean of the inverses given in A001519
and A001906.-David W.Cantrell (DWCantrell(AT)sigmaxi.net),Feb
19 2007
%C A000045 Comment from Larry Gerstein (gerstein(AT)math.ucsb.edu),Mar
30 2007:A result of
Jacobi from 1848 states that every symmetric matrix over a p.i.d.is
congruent to a triple-diagonal matrix.Consider the maximal number
T(n) of summands in the determinant of an n X n triple-diagonal
matrix.This is the same as the number of summands in such a
determinant
in which the main-,sub-,and super-diagonal elements are all nonzero.By
expanding on the first row we see that the sequence of T(n)'s
is the Fibonacci sequence without the initial stammer on the 1's.%C
A000045 Suppose psi=ln(phi).We get the representation F(n)=(2/sqrt(5))
*sinh(n*psi) if n
is even;F(n)=(2/sqrt(5))*cosh(n*psi) if n is odd.There is a similar
representation for Lucas numbers (A000032).Many Fibonacci formulas
now easily follow from appropriate sinh-and cosh-formulas.For
example:the de Moivre theorem (cosh(x)+sinh(x))^m=cosh(mx)+sinh(mx)
produces
L(n)^2+5F(n)^2=2L(2n) and L(n)F(n)=F(2n) (setting x=n*psi and m=2).-
Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de),Apr 18 2007
%C A000045 Inverse:floor(log_phi(sqr(5)*Fib(n))+0.5)=n,for n>1. Also
for n>0,floor(1/2*log_phi(5*Fib(n)*Fib(n+1)))=n.Extension valid for
integer n,except n=0,-1:floor(1/2*sign(Fib(n)*Fib(n+1))*log_phi|5*Fib
(n)*Fib(n+1)|)=n {where sign(x)=sign of x}.-Hieronymus Fischer
(Hieronymus.Fischer(AT)gmx.de),May 02 2007
%C A000045 F(n+2)=The number of Khalimsky-continuous functions with a
two-point codomain.-Shiva Samieinia (shiva(AT)math.su.se),Oct 04 2007
%C A000045 From Kauffman and Lopes,Proposition 8.2,p.21:"The sequence
of the determinants
of the Fibonacci sequence of rational knots is the
Fibonacci sequence
(of numbers)."-Jonathan Vos Post (jvospost3(AT)
gmail.com),Oct
26 2007
%C A000045 This is a_1(n) in the Doroslovacki reference.%C A000045 Let
phi=1.6180339...;then phi^n=(1/phi)*a(n)+a(n
+1).Example:phi^4=6.8541019...=(.6180339...)*3+5. Also phi=1/1+1/2+1/
(2*5)+1/(5*13)+1/(13*34)+1/(34*89),...-Gary W.Adamson (qntmpkt(AT)
yahoo.com),Dec 15 2007
%C A000045 The sequence of first differences,fib(n+1)-fib(n),is
essentailly the same sequence:1,0,1,1,2,3,5,8,13,21,34,55,89,144,...-
Colm Mulcahy,Mar 03 2008
%C A000045 a(n)=the number of different ways to run up a staircase
with n steps,taking steps
of odd sizes where the order is relevant and there is no other
restriction
on the number or the size of each step taken.-Mohammad K.Azarian
(azarian(AT)evansville.edu),May 21 2008
%C A000045 Equals row sums of triangle A144152.[From Gary W.Adamson
(qntmpkt(AT)yahoo.com),Sep 12 2008]
%C A000045 Contribution from Cino Hilliard (hillcino368(AT)
gmail.com),Sep 15 2008):(Start)
%C A000045 Except for the initial term,the numerator of the
convergents to the recursion x
%C A000045=1/(x+1).(End)
%C A000045 Contribution from Ross Drewe (rd(AT)labyrinth.net.au),Oct
05 2008):(Start)
%C A000045 F(n) is the number of possible binary sequences of length n
that obey the
%C A000045 sequential construction rule:if last symbol is 0,add the
complement (1);
%C A000045 else add 0 or 1. Here 0,1 are metasymbols for any 2-valued
symbol set.This
%C A000045 rule has obvious similarities to JFJ Laros's rule,but is
based on addition
%C A000045 rather than substitution,and creates a tree rather than a
single sequence.(End)
%D A000045 P.Bachmann,Niedere Zahlentheorie (1902,1910),reprinted
Chelsea,NY,1968,vol.2,p.70.
%D A000045 R.B.Banks,Slicing Pizzas,Racing Turtles,and Further
Adventues in Applied Mathematics,Princeton Univ.Press,1999. See p.84.
%D A000045 Paul Barry,A Catalan Transform and Related Transformations
on Integer Sequences,Journal of Integer Sequences,Vol.8 (2005),Article
05.4.5.%D A000045 Paul Barry,On Integer-Sequence-Based Constructions
of Generalized Pascal Triangles,Journal of Integer Sequences,Vol.9
(2006),Article 06.2.4.%D A000045 A.T.Benjamin and J.J.Quinn,Proofs
that really count:the art of combinatorial
proof,M.A.A.2003,id.4.
%D A000045 Majorie Bicknell and Verner E Hoggatt,Fibonacci's Problem
Book,Fibonacci Association,San Jose,Calif.,1974.
%D A000045 S.Brlek,E.Duchi,E.Pergola and S.Rinaldi,On the equivalence
problem for succession
rules,Discr.Math.,298 (2005),142-154.
%D A000045 Hongwei Chen,Evaluations of Some Variant Euler Sums,Journal
of Integer Sequences,Vol.9 (2006),Article 06.2.3.%D A000045 B.A.Davey
and H.A.Priestley,Introduction to Lattices and Order (2nd edition),CUP,
2002. (See Exercise 1.15.)
%D A000045 B.Davis,'The law of first digits' in'Science
Today'(subsequently renamed'2001')March
1980 pp.55,Times of India,Mumbai.%D A000045 Nathaniel D.Emerson,A
Family of Meta-Fibonacci Sequences Defined by Variable-Order
Recursions,Journal of Integer Sequences,Vol.9 (2006),Article
06.1.8.%D A000045 Reinhardt Euler,The Fibonacci Number of a Grid Graph
and a New Class of Integer
Sequences,Journal of Integer Sequences,Vol.8 (2005),Article 05.2.6.%D
A000045 G.Everest,A.van der Poorten,I.Shparlinski and
T.Ward,Recurrence Sequences,Amer.Math.Soc.,2003;see esp.p.255.
%D A000045 Emmanuel Ferrand,Deformations of the Taylor Formula,Journal
of Integer Sequences,Vol.10 (2007),Article 07.1.7.%D A000045
S.R.Finch,Mathematical Constants,Cambridge,2003,Section 1.2.%D A000045
R.P.Grimaldi,Compositions without the summand 1,Proceedings Thirty-
second Southeastern
International Conference on Combinatorics,Graph Theory and Computing
(Baton Rouge,LA,2001).Congr.Numer.152 (2001),33-43.
%D A000045 N.S.S.Gu,N.Y.Li and T.Mansour,2-Binary trees:bijections and
related issues,Discr.Math.,308 (2008),1209-1221.
%D A000045 G.H.Hardy and E.M.Wright,An Introduction to the Theory of
Numbers.3rd ed.,Oxford Univ.Press,1954;see esp.p.148.
%D A000045 J.Hermes,Anzahl der Zerlegungen einer ganzen rationalen
Zahl in Summanden,Math.Ann.,45 (1894),371-380.
%D A000045 V.E.Hoggatt,Jr.,Fibonacci and Lucas
Numbers.Houghton,Boston,MA,1969.
%D A000045 E.Horowitz and S.Sahni,Fundamentals of Data
Structures,Computer Science Press,1976;p.338.
%D A000045 C.W.Huegy and D.B.West,A Fibonacci tiling of the
plane,Discrete Math.,249
(2002),111-116.
%D A000045 D.E.Knuth,The Art of Computer Programming.Addison-
Wesley,Reading,MA,Vol.1,p.78;Vol.3,Section 6.2.1.%D A000045 Thomas
Koshy,"Fibonacci and Lucas Numbers with Applications",John Wiley and
Sons,2001.
%D A000045 Lukovits et al.,Nanotubes:Number of Kekule structures and
aromaticity,J.Chem.Inf.Comput.Sci,(2003),vol.43,609-614. See eq.2 on
page 610.
%D A000045 I.Lukovits and D.Janezic,"Enumeration of conjugated
circuits in nanotubes",J.Chem.Inf.Comput.Sci.,vol.44,410-414
(2004).See Table 1 second
column.%D A000045 B.Malesevic:Some combinatorial aspects of
differential operation composition on
the space R^n,Univ.Beograd,Publ.Elektrotehn.Fak.,Ser.Mat.9 (1998),
29-33.
%D A000045 Tony D.Noe and Jonathan Vos Post,Primes in Fibonacci n-step
and Lucas n-step Sequences,Journal of Integer Sequences,Vol.8
(2005),Article 05.4.4.%D A000045 P.Ribenboim,The New Book of Prime
Number Records,Springer,1996.
%D A000045 J.Riordan,An Introduction to Combinatorial
Analysis,Princeton University Press,Princeton,NJ,1978.
%D A000045 A.M.Robert,A Course in p-adic Analysis,Springer-Verlag,
2000;p.213.
%D A000045 J.Roberts,Lure of the Integers,Math.Assoc.America,1992,p.
288.
%D A000045 A.Sapounakis,I.Tasoulas and P.Tsikouras,On the Dominance
Partial Ordering of
Dyck Paths,Journal of Integer Sequences,Vol.9 (2006),Article
06.2.5.%D A000045 Mark A.Shattuck and Carl G.Wagner,Periodicity and
Parity Theorems for a Statistic
on r-Mino Arrangements,Journal of Integer Sequences,Vol.9
(2006),Article 06.3.6.%D A000045 Michael Z.Spivey and Laura
L.Steil,The k-Binomial Transforms and the Hankel Transform,Journal of
Integer Sequences,Vol.9 (2006),Article 06.1.1.%D A000045
S.Vajda,Fibonacci and Lucas numbers,and the Golden Section,Ellis
Horwood Ltd.,Chichester,1989.
%D A000045 N.N.Vorob'ev,Chisla fibonachchi[Russian],Moscow,1951.
English translation,Fibonacci Numbers,Blaisdell,New York and London,
1961.
%D A000045 N.N.Vorobiev,Fibonacci Numbers,Birkhauser (Basel;Boston)
2002.
%D A000045 D.Wells,The Penguin Dictionary of Curious and Interesting
Numbers,pp.61-7,Penguin
Books 1987.
%D A000045 Tianping Zhang and Yuankui Ma,On Generalized Fibonacci
Polynomials and Bernoulli
Numbers,Journal of Integer Sequences,Vol.8 (2005),Article 05.5.3.%D
A000045 Clifford A.Pickover,A Passion for Mathematics,Wiley,2005;see p.
49.
%D A000045 Mohammad K.Azarian,The Generating Function for the
Fibonacci Sequence,Missouri
Journal of Mathematical Sciences,Vol.2,No.2,Spring 1990,pp.78-79.
Zentralblatt MATH,Zbl 1097.11516.%D A000045 Mohammad K.Azarian,A
Generalization of the Climbing Stairs Problem II,Missouri
Journal of Mathematical Sciences,Vol.16,No.1,Winter 2004,pp.12-17.
%D A000045 A.Milicevic and N.Trinajstic,"Combinatorial Enumeration in
Chemistry",Chem.Modell.,Vol.4,(2006),pp.405-469.
%D A000045 A.S.Posamentier& I.Lehmann,The Fabulous Fibonacci
Numbers,Prometheus Books,Amherst,NY 2007.
%D A000045 Aimei Yu and Xuezheng Lv,"The Merrifield-Simmons indices
and Hosoya indices of
trees with k pendant vertices.",J.Math.Chem.,Vol.41
(2007),pp.33-43. See page 35.-from Parthasarathy Nambi (PachaNambi(AT)
yahoo.com),Sep 01 2008
%H A000045 N.J.A.Sloane, <a href="http://www.research.att.com/~njas/
sequences/b000045.txt">The first 500 Fibonacci numbers:Table of n,F(n)
for n=0..500</a>%H A000045 Joerg Arndt, <a href="http://www.jjj.de/fxt/
#fxtbook">Fxtbook</a>%H A000045 Amazing Mathematical Object Factory,
<a href="http://www.schoolnet.ca/vp-pv/amof/
e_fiboI.htm">Information on the Fibonacci sequences</a>
%H A000045 M.Anderson et al., <a href="http://library.thinkquest.org/
27890/theSeries.html">The Fibonacci Series</a>%H A000045 Matt
Anderson,Jeffrey Frazier,and Kris Popendorf, <a href="http://
library.thinkquest.org/
27890/theSeries4.html">The Fibonacci series:the
successor formula</a>%H A000045 Matt Anderson,Jeffrey Frazier,and Kris
Popendorf,<a href="http://library.thinkquest.org/
27890/theSeries.html">The Fibonacci series:the section
index</a>%H A000045 P.G.Anderson, <a href="http://www.cs.rit.edu/~pga/
Fibo/fact_***.html">Fibonacci
Facts</a>%H A000045 H.Bottomley and N.J.A.Sloane, <a href="http://
www.research.att.com/~njas/sequences/
a45.html">Illustration of initial terms:the Fibonacci
tree</a>%H A000045 M.Boulanger, <a href="http://www.easymaths.org/
fibonacci1.html">Rabbit Puzzle</a>[Broken link?]
%H A000045 Brantacan, <a href="http://www.branta/connectfree.co.uk/
fibonacci.htm">Fibonacci
Numbers</a>%H A000045 J.Britton& B.V.Eeckhout, <a href="http://
ccins.camosun.bc.ca/~jbritton/fibonacci/
jbfibapplet.htm">Fibonacci Interactive</a>%H A000045
C.K.Caldwell, <a href="http://primes.utm.edu/glossary/page.php?
sort=FibonacciNumber">Fibonacci Numbers</a>%H A000045 C.K.Caldwell,The
Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/
FibonacciNumber.html">Fibonacci number</a>%H A000045
P.J.Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Sequences
realized by oligomorphic permutation groups</a>,J.Integ.Seqs.Vol.3
(2000),#00.1.5.%H A000045 C.Conner, <a href="http://www.geocities.com/
cyd-conner/page1.html">Fibonacci</a>%H A000045 E.S.Croot, <a
href="http://www.math.gatech.edu/~ecroot/recurrence_notes2.pdf";>Notes
on Linear Recurrence Sequences</a>%H A000045 C.Dement, <a href="http://
mathforum.org/discuss/sci.math/t/622432">Posting to Math
Forum</a>.%H A000045 R.M.Dickau, <a href="http://mathforum.org/
advanced/robertd/fibboard.html">Fibonacci
numbers</a>%H A000045 R.Doroslovacki, <a href="http://www.emis.de/
journals/MV/9434/mv943407.ps">Binary
sequences without 011...110 (k-1 1's) for fixed k</a>,Mat.Vesnik
46 (1994),no.3-4,93-98.
%H A000045 Enthios LLC, <a href="http://www.enthios.com/
FibonacciPrimer.htm">Fibonacci Primer</a>%H A000045 G.Everest,A.J.van
der Poorten,Y.Puri and T.Ward, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Integer Sequences and Periodic
Points</a>
,Journal of Integer Sequences,Vol.5 (2002),Article 02.2.3
%H A000045 D.Foata and G.-N.Han, <a href="http://www-irma.u-strasbg.fr/
~foata/paper/pub71.html">Nombres de Fibonacci et polynomes
orthogonaux</a>,%H A000045 I.Galkin, <a href="http://ulcar.uml.edu/
~iag/CS/Fibonacci.html">"Fibonacci Numbers
Spelled Out"</a>%H A000045 L.Goldsmith, <a href="http://
people.bath.ac.uk/ma2lag/fibonaccinumbers.html">The
Fibonacci Numbers</a>%H A000045 A.P.Hillman& G.L.Alexanderson,Algebra
Through Problem Solving,Chapter 2 pp.11-16, <a href="http://
education.lanl.gov/RESOURCES/ATPS/CHPTR02/
P011.HTM">The Fibonacci and Lucas Numbers</a>%H A000045
INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/
encyclopedia?Search=ECSnb&argsearch=9">Encyclopedia of Combinatorial
Structures 9</a>%H A000045 R.Javonovic, <a href="http://
milan.milanovic.org/math/english/function/function.html">Fibonacci
Function Calculator</a>%H A000045 R.Javonovic, <a href="http://
milan.milanovic.org/math/english/pdf/Fibonacci.pdf">The relations
between the Fibonacci and the Lucas numbers</a>%H A000045 R.Jovanovic,
<a href="http://milan.milanovic.org/math/Math.php?
akcija=SviFibo">First 70 Fibonacci numbers</a>%H A000045 S.Kak, <a
href="http://uk.arXiv.org/abs/physics/0411195";>The Golden Mean and the
Physics of Aesthetics</a>%H A000045 Louis H.Kauffman and Pedro Lopes,
<a href="http://arXiv.org/pdf/0710.3765";>Graded
forests and rational knots</a>,Oct 19,2007.
%H A000045 B.Kelly, <a href="http://home.att.net/~blair.kelly/
mathematics/fibonacci/">Fibonacci
and Lucas factorizations</a>%H A000045 Tanya Khovanova, <a
href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>%H A000045
C.Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Matrix
Transformations of Integer Sequences</a>,J.Integer Seqs.,Vol.6,2003.
%H A000045 R.Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/
R.Knott/Fibonacci/">Fibonacci
numbers and the golden section</a>%H A000045 R.Knott, <a href="http://
www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html">Mathematics
of the Fibonacci Series</a>%H A000045 R.Knott, <a href="http://
www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html">Fibonacci
numbers with tables of F(0)-F(500).</a>%H A000045
A.Krowne,PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
FibonacciNumber.html">Fibonacci sequence</a>%H A000045 A.F.Labossiere,
<a href="http://members.lycos.co.uk/sobalian/index.html";>Sobalian
Coefficients</a>.%H A000045 A.F.Labossiere, <a href="http://
members.lycos.co.uk/stereotomography/index.html">Miscellaneous</a>.%H
A000045 M.A.Lerma, <a href="http://www.math.northwestern.edu/~mlerma/
problem_solving/results/
recurrences.pdf">Recurrence Relations</a>%H A000045
D.Litchfield,D.Goldenheim and C.H.Dietrich, <a href="http://
scientium.com/diagon_alley/
archival/segments/euclid1.htm">Euclid,Fibonacci and
Sketchpad</a>
,Math.Teacher,90 (1997).%H A000045 B.Malesevic, <a href="http://
matematika.etf.bg.ac.yu/publikacije/pub/P09(98)/P09_06.ZIP">Some
combinatorial aspects of differential operation composition
on the space R^n</a>.%H A000045 D.Merlini,R.Sprugnoli and M.C.Verri,
<a href="http://www.dsi.unifi.it/~merlini/
tiling.ps">Strip tiling and regular grammars</
a>,Theoret.Computer
Sci.242,1-2 (2000) 109-124.
%H A000045 D.Merrill, <a href="http://pw1.netcom.com/~merrills/
fibphi.html">The Fib-Phi Link
Page</a>%H A000045 Jean-Christophe Michel, <a href="http://
framy.free.fr/fibonacci%20dans%20mandelbrot.htm">Le nombre d'or dans
l'ensemble de Mandelbrot</a>(in French,'The golden number in the
Mandelbrot set')
%H A000045 H.Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/
mathland/matha1/matha108.htm">Factorizations of many number sequences</
a>%H A000045 H.Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/
mathland/matha1/matha109.htm">Factorizations of many number sequences</
a>%H A000045 H.Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/
mathland/matha1/matha110.htm">Factorizations of many number sequences</
a>%H A000045 H.Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/
mathland/matha1/matha111.htm">Factorizations of many number sequences</
a>%H A000045 H.Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/
mathland/matha1/matha112.htm">Factorizations of many number sequences</
a>%H A000045 P.Moree, <a href="http://arXiv.org/abs/math.CO/
0311205">Convoluted convolved Fibonacci
numbers</a>%H A000045 Newton's Institute, <a href="http://
www.newton.cam.ac.uk/wmy2kposters/january">Posters
in the London Underground</a>%H A000045 J.Patterson, <a href="http://
www.bath.ac.uk/~ma1jmp/link.html">The Fibonacci Sequence</a>%H A000045
Ivars Peterson, <a href="http://www.sciencenews.org/articles/20060603/
mathtrek.asp">Fibonacci's Missing Flowers</a>.%H A000045
S.Plouffe,Project Gutenberg, <a href="http://ibiblio.org/pub/docs/
books/gutenberg/
etext01/fbncc10.txt">The First 1001 Fibonacci Numbers</
a>%H A000045 S.Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/
OEIS/A000045">Fibonacci numbers</a>[Contains the first 754965 terms]
%H A000045 N.Renton, <a href="http://www.users.bigpond.net.au/renton/
903.htm">The fibonacci
Series</a>%H A000045 B.Rittaud, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/VOL10/Rittaud2/rittaud11.pdf">On the Average Growth of
Random Fibonacci Sequences</a>,Journal
of Integer Sequences,10 (2007),Article 07.2.4.%H A000045 E.S.Rowland,
<a href="http://www.math.rutgers.edu/~erowland/
fibonacci.html">Fibonacci
Sequence Calculator up to n=1474</a>%H A000045 Shiva Samieinia, <a
href="http://www.math.su.se/reports/2007/6/";>Digital straight
line segments and curves</a>.Licentiate Thesis.Stockholm
University,Department of Mathematics,Report 2007:6.
%H A000045 D.Schweizer, <a href="http://math.holycross.edu/~davids/
fibonacci/fibonacci.html">First 500 Fibonacci Numbers in blocks of
100.</a>%H A000045 S.Silvia, <a href="http://arttech.about.com/library/
weekly/aa060900a_fibonacci_sequence.htm">Fibonacci sequence</a>%H
A000045 Jaap Spies, <a href="http://www.jaapspies.nl/oeis/
a000045.sage">SAGE program for
computing A000045</a>%H A000045 Z.-H.Sun, <a href="http://
202.195.112.2/xsjl/szh/ConFn.pdf">Congruences For Fibonacci
Numbers</a>%H A000045 Roberto Tauraso, <a href="http://
www.cs.uwaterloo.ca/journals/JIS/">A New Domino
Tiling Sequence</a>,Journal of Integer Sequences,Vol.7 (2004),Article
04.2.3.%H A000045 Thesaurus.Maths.org, <a href="http://
thesaurus.maths.org/dictionary/map/word/3788">Fibonacci sequence</a>%H
A000045 K.Tognetti, <a href="http://www.austms.org.au/Modules/
Fib">Fibonacci-His Rabbits
and His Numbers and Kepler</a>%H A000045 Carl G.Wagner, <a
href="http://www.cs.uwaterloo.ca/journals/JIS/index.html";>Partition
Statistics and q-Bell Numbers (q=-1)</a>,J.Integer Seqs.,Vol.7,2004.
%H A000045 N.P.Watson, <a href="http://www.hjnpwatson.demon.co.uk/
javafibn.htm">First 50
Fibonacci Numbers</a>%H A000045 Eric Weisstein's World of Mathematics,
<a href="http://mathworld.wolfram.com/FibonacciNumber.html";>Link to a
section of The World of Mathematics.</a>%H A000045 Eric Weisstein's
World of Mathematics, <a href="http://mathworld.wolfram.com/Double-
FreeSet.html">Link to a section of The World of Mathematics.</a>%H
A000045 Eric Weisstein's World of Mathematics, <a href="http://
mathworld.wolfram.com/Fibonaccin-StepNumber.html">Link to a section of
The World of Mathematics.</a>%H A000045 Eric Weisstein's World of
Mathematics, <a href="http://mathworld.wolfram.com/
ResistorNetwork.html">Link to a section of The World of Mathematics.</
a>%H A000045 Wikipedia, <a href="http://www.wikipedia.org/wiki/
Fibonacci_number">Fibonacci number</a>%H A000045 Willem's Fibonacci
site, <a href="http://home.zonnet.nl/LeonardEuler/
fiboe.htm">Fibonacci</a>%H A000045 G.Xiao,Numerical Calculator, <a
href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html";>To
display F(n),for n up to 78365,operate on "fibonacci(n)"</a>%H
A000045<a href="http://www.research.att.com/~njas/sequences/
Sindx_Cor.html#core">Index
entries for "core" sequences</a>%H A000045<a href="http://
www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index
entries for related partition-counting sequences</a>%H A000045
S.Rabinowitz, <a href="http://www.mathpropress.com/stan/bibliography/
algorithmicFib.pdf">Algorithmic Manipulation of Fibonacci Identities</
a>(1996).[From R.J.Mathar (mathar(AT)strw.leidenuniv.nl),Nov 06 2008]
%H A000045 N.N.Vorob'ev, <a href="http://eom.springer.de/F/
f040020.htm">Fibonacci numbers</a>,Springer's Encyclopaedia of
Mathematics.[From R.J.Mathar (mathar(AT)strw.leidenuniv.nl),Nov 06
2008]
%F A000045 G.f.:x/(1-x-x^2).%F A000045 F(n)=((1+sqrt(5))^n-(1-sqrt(5))
^n)/(2^n*sqrt(5)).%F A000045 Alternatively,F(n)=((1/2+sqrt(5)/2)^n-
(1/2-sqrt(5)/2)^n)/sqrt(5).%F A000045 F(n)=F(n-1)+F(n-2)=-(-1)^n F(-n).
%F A000045 F(n)=round(phi^n/sqrt(5)).%F A000045 F(n+1)=Sum(0<=j<=[n/
2];binomial(n-j,j))
%F A000045 E.g.f.:(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2).-Len Smiley
(smiley(AT)math.uaa.alaska.edu),Nov 30 2001
%F A000045[0 1;1 1]^n[0 1]=[F(n);F(n+1)]
%F A000045 x|F(n)==>x|F(kn).%F A000045 A sufficient condition for F(m)
to be divisible by a prime p is (p-1) divides
m,if p==1 or 4 (mod 5);(p+1) divides m,if p==2 or 3 (mod 5);or 5
divides m,if p=5. (This is essentially Theorem 180 in Hardy and
Wright.)-Fred W.Helenius (fredh(AT)ix.netcom.com),Jun
29,2001
%F A000045 a(n)=F(n) has the property:F(n)*F(m)+F(n+1)*F(m+1)=F(n+m+1)-
Miklos Kristof
(kristmikl(AT)freemail.hu),Nov 13 2003
%F A000045 Kurmang.Aziz.Rashid (Kurmang.Rashid(AT)Btopenworld.com),Feb
21 2004,makes 4
conjectures and gives 3 theorems:%F A000045 Conjecture 1:for n>=2 sqrt
{F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4)+2*(-1)^n}={F(2n+1)+2*(-1)^n}/F
(n-1).Conjecture 2:for n>=0,{F(n+2)*F(n+3)}-{F(n+1)*F(n+4)}+(-1)^n=0.
%F A000045 Conjecture 3:for n>=0,F(2n+1)^3-F(2n+1)*[(2*A^2)-1]-[A+A^3]
=0,where A={F(2n+1)+sqrt{5*F(2n+1)^2+4}}/2
%F A000045 Conjecture 4:for x>=5,if x is a Fibonacci number>=5 then
g*x*[{x+sqrt{5*(x^2)+-4}}/2]*[2x+{{x+sqrt{5*(x^2)+-4}}/2}]*[2x+{{3x
+3*sqrt {5*(x^2)+-4}}/2}]^2+[2x+{{x+sqrt{5*(x^2)+-4}}/2}]+-x*[2x+{{3x
+3*sqrt{5*(x^2)+-4}}/2}]^2-x*[2x+{{x+sqrt{5*(x^2)+-4}}/2}]*[x+{{x+sqrt
{5*(x^2)+-4}}/2}]*[2x+{{3x+3*sqrt{5*(x^2)+-4}}/2}]^2=0,where g={1+sqrt
5/2}.%F A000045 Theorem 1:for n>=0,{F(n+3)^2-F(n+1)^2}/F(n+2)={F(n+3)+F
(n+1)}.Theorem 2:for n>=0,F(n+10)=11*F(n+5)+F(n).Theorem 3:for n>=6,F
(n)
=4*F(n-3)+F(n-6).%F A000045 Conjecture 2 of Rashid is
actually a special case of the general law F(n)*F(m)+F(n+1)*F(m+1)=F(n
+m+1) (take n<-n+1 and m<--(n+4) in this law).-Harmel Nestra
(harmel.nestra(AT)ut.ee),Apr 22 2005
%F A000045 Conjecture:for all c such that 2-Phi<=c<2*(2-Phi) we have F
(n)=floor(Phi*a(n-1)+c)
for n>2-Gerald McGarvey (Gerald.McGarvey(AT)comcast.net),Jul
21 2004
%F A000045|2*Fib(n)-9*Fib(n+1)| =4*A000032(n)+A000032(n+1).-Creighton
Dement (creighton.k.dement(AT)uni-oldenburg.de),Aug 13 2004
%F A000045 For x>Phi,Sum n=0..inf F(n)/x^n=x/(x^2-x-1)-Gerald McGarvey
(gerald.mcgarvey(AT)comcast.net),Oct 27 2004
%F A000045 F(n+1)=exponent of the n-th term in the series f(x,1)
determined by the equation
f(x,y)=xy+f(xy,x).-Jonathan Sondow (jsondow(AT)
alumni.princeton.edu),Dec 19 2004
%F A000045 a(n-1)=sum(k=0,n,(-1)^k*binomial(n-ceil(k/2),floor(k/2)))-
Benoit Cloitre (benoit7848c(AT)orange.fr),May 05 2005
%F A000045 F(n+1)=sum{k=0..n,binomial((n+k)/2,(n-k)/2)(1+(-1)^(n-k))/
2};-Paul Barry (pbarry(AT)wit.ie),Aug 28 2005
%F A000045 Fibonacci(n)=Product(1+4[cos(j*Pi/n)]^2,j=1..ceil(n/2)-1).
[Bicknell and Hoggatt,pp.47-48]-Emeric Deutsch,Oct 15 2006
%F A000045 F(n)=2^-(n-1)*sum{k=0..floor((n-1)/2),binomial(n,2*k+1)
*5^k};-Hieronymus Fischer
(Hieronymus.Fischer(AT)gmx.de),Feb 07 2006
%F A000045 a(n)=(b(n+1)+b(n-1))/n where {b(n)} is the sequence A001629-
Sergio Falcon (sfalcon(AT)dma.ulpgc.es),Nov 22 2006
%F A000045 F(n*m)=Sum{k=0..m,binomial(m,k)*F(n-1)^k*F(n)^(m-k)*F(m-
k)}.The generating
function of F(n*m) (n fixed,m=0,1,2...) is G(x)=F(n)*x/((1-F (n-1)*x)
^2-F(n)*x*(1-F(n-1)*x)-(F(n)*x)^2).E.g.F(15)=610=F(5*3)
=binomial(3,0)*F(4)^0*F(5)^3*F(3)+binomial(3,1)*F(4)^1*F
(5)^2*F(2)
+binomial(3,2)*F(4)^2*F(5)^1*F(1)+binomial(3,3)*F(4)^3*F(5)^0*F(0)
=1*1*125*2+3*3*25*1+3*9*5*1+1*27*1*0=250+225+135+0
=610-Miklos Kristof,Feb 12 2007
%F A000045 Comments from Miklos Kristof (kristmikl@xxxxxxxxxxx),Mar 19
2007 (Start)
%F A000045 Let L(n)=A000032=Lucas numbers.Then:%F A000045 For a>=b and
odd b,F(a+b)+F(a-b)=L(a)*F(b).%F A000045 For a>=b and even b,F(a+b)+F
(a-b)=F(a)*L(b).%F A000045 For a>=b and odd b,F(a+b)-F(a-b)=F(a)*L(b)..
%F A000045 For a>=b and even b,F(a+b)-F(a-b)=L(a)*F(b).%F A000045 F(n
+m)+(-1)^m*F(n-m)=F(n)*L(m);
%F A000045 F(n+m)-(-1)^m*F(n-m)=L(n)*F(m);
%F A000045 F(n+m+k)+(-1)^k*F(n+m-k)+(-1)^m*(F(n-m+k)+(-1)^k*F(n-m-k))=F
(n)*L(m)*L(k);
%F A000045 F(n+m+k)-(-1)^k*F(n+m-k)+(-1)^m*(F(n-m+k)-(-1)^k*F(n-m-k))=L
(n)*L(m)*F(k);
%F A000045 F(n+m+k)+(-1)^k*F(n+m-k)-(-1)^m*(F(n-m+k)+(-1)^k*F(n-m-k))=L
(n)*F(m)*L(k);
%F A000045 F(n+m+k)-(-1)^k*F(n+m-k)-(-1)^m*(F(n-m+k)-(-1)^k*F(n-m-k))
=5*F(n)*F(m)*F(k).(End)
%F A000045 Fib(n)=b(n)+(p-1)*sum{1<k<n,floor(b(k)/p)*Fib(n-k+1)} where
b(k) is the digital
sum analogue of the Fibonacci recurrence,defined by b(k)=ds_p(b(k-1))
+ds_p(b(k-2)),b(0)=0,b(1)=1,ds_p=digital sum base p.Example for base
p=10:Fib(n)=A010077(n)+9*sum{1<k<n,A059995(A010077(k))*Fib(n-k+1)}.-
Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de),Jul 01 2007
%F A000045 Fib(n)=b(n)+p*sum{1<k<n,floor(b(k)/p)*Fib(n-k+1)} where b
(k) is the digital product
analogue of the Fibonacci recurrence,defined by b(k)=dp_p(b(k-1))+dp_p
(b(k-2)),b(0)=0,b(1)=1,dp_p=digital product base p.Example for base
p=10:Fib(n)=A074867(n)+10*sum{1<k<n,A059995(A074867(k))*Fib(n-k+1)}.-
Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de),Jul 01 2007
%F A000045 a(n)=denominator of continued fraction[1,1,1,...],(with n
ones);e.g.2/3=continued fraction[1,1,1];where barover[1]=[1,1,1...]=.
6180339,...-Gary W.Adamson (qntmpkt(AT)yahoo.com),Nov 29 2007
%F A000045 F(n+3)=2F(n+2)-F(n),F(n+4)=3F(n+2)-F(n),F(n+8)=7F(n+4)-F
(n),F(n+12)=18F(n+6)-F(n).-Paul Curtz (bpcrtz(AT)free.fr),Feb 01 2008
%F A000045 1=1/(1*2)+1/(1*3)+1/(2*5)+1/(3*8)+1/
(5*13)+...=1/2+1/3+1/10+1/24+1/65+1/168+...;where A059929=
(0,2,3,10,24,65,168,...).-Gary W.Adamson (qntmpkt(AT)yahoo.com),Mar 16
2008
%F A000045 a(2^n)=prod{i=0}^{n-2}B(i) where B(i) is A001566.Example
3*7*47=Fib(16)-Kenneth
J Ramsey (Ramsey2879(AT)msn.com),Apr 23 2008
%F A000045 F(n)=(1/(n-1)!)*[n^(n-1)-{C(n-2,0)+4*C(n-2,1)+3*C(n-2,2)}*n^
(n-2)+{10*C(n-3,0)+49*C(n-3,1)+95*C(n-3,2)+83*C(n-3,3)+27*C(n-3,4)}*n^
(n-3)-{90*C(n-4,0)+740*C(n-4,1)+2415*C(n-4,2)+4110*C(n-4,3)+3890*C
(n-4,4)+1950*C(n-4,5)+405*C(n-4,6)}*n^(n-4)+.....].-Andre F.Labossiere
(boronali(AT)laposte.net),Nov 24 2004
%F A000045 a(n+1)=Sum_{k,0<=k<=n} A109466(n,k)*(-1)^(n-k).[From
Philippe DELEHAM (kolotoko(AT)wanadoo.fr),Oct 26 2008]
%e A000045 Contribution from Cino Hilliard (hillcino368(AT)
gmail.com),Sep 15 2008):(Start)
%e A000045 For x=0,1,2,3,4 x=1/(x+1)=1,1/2,2/3,3/5,5/8,These fractions
have
%e A000045 numerators 1,1,2,3,5 the 2nd to 6-th entries in the
sequence.(End)
%p A000045 with(combinat):A000045:=proc(n) fibonacci(n);end;
%p A000045 ZL:=[S,{a=Atom,b=Atom,S=Prod(X,Sequence(Prod
(X,b))),X=Sequence(b,card>=1)},unlabelled]:seq(combstruct[count]
(ZL,size=n),n=0..38);
-Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),Apr 04 2008
%p A000045 spec:=[B,{B=Sequence(Set(Z,card>1))},unlabeled]:seq
(combstruct[count](spec,size=n),n=1..39);-Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com),Apr 04 2008
%t A000045 Table[Fibonacci[k],{k,1,50}]
%o A000045 (AXIOM)[fibonacci(n) for n in 0..50]
%o A000045 (MAGMA) F:=func<n|Fibonacci(n)>;
%o A000045 (PARI) a(n)=fibonacci(n)
%o A000045 (PARI) a(n)=imag(quadgen(5)^n)
%o A000045 (PARI) a(n)=if(n<0,-(-1)^n*a(-n),if(n<2,n,a(n-1)+a(n-2)))
%o A000045 # Python program from Jaap Spies,Jan 05,2007 (Change
leading dots to blanks.)
%o A000045 def fib():%o A000045... """
%o A000045 ....... generates an "infinity" of Fibonacci numbers,
%o A000045 ....... starting with 1
%o A000045 ... """
%o A000045... x,y=0,1
%o A000045... while 1:%o A000045.......x,y=y,x+y
%o A000045.......yield x
%o A000045................%o A000045 f=fib()
%o A000045 a=[f.next() for i in range(1000)] # 1000 or more
%o A000045 a.insert(0,0)
%o A000045................%o A000045 def A000045(n):%o A000045... """
returns Fibonacci number with index n, offset 0,4 """
%o A000045... return a[n]
%o A000045................%o A000045 def A000045_list(N):%o A000045...
""" returns a list of the first n Fibonacci numbers """
%o A000045... return a[:N]
%o A000045................%o A000045 # (SAGE) Demonstration program
from Jaap Spies:%o A000045 # To see which functions are available
type:sloane.A[tab]
%o A000045 # All builtin SAGE programs are called the same way:%o
A000045 # a=sloane.A000045;a # This returns the name of the sequence
%o A000045 # a(n) # This returns the n-th number of the sequence:%o
A000045 # a.list(n) # This returns a list of the first n numbers:%o
A000045 # Copy and paste the following into a work*** or the
interpreter:%o A000045 a=sloane.A000045;print a
%o A000045 print a(0)
%o A000045 print a(1)
%o A000045 print a(2)
%o A000045 print a(38)
%o A000045 print a.list(39)
%o A000045 sage:from sage.combinat.sloane_functions import recur_gen2
sage:it=recur_gen2(0,1,1,1) sage:[it.next() for i in range(30)]-
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),Jun 25 2008
%o A000045 (PARI) x=0;for(j=0,100,x=1/(x+1);print1(numerator(x)","))
[From Cino Hilliard (hillcino368(AT)gmail.com),Sep 15 2008]
%Y A000045 Cf.A039834 (signed Fibonacci numbers).%Y A000045
Cf.A000213,A000288,A000322,A000383,A060455,A030186,A039834,A020695,A020701,A071679.%Y
A000045 Cf.A099731,A100492,A094216,A094638,A000108,A101399,A101400.%Y
A000045 First row of array A103323.Second row of array A099390.%Y
A000045 Row 2 of arrays A048887 and A092921 (k-generalized Fibonacci
numbers).%Y A000045 a(n)=A094718(4,n).a(n)=A101220(0,j,n).%Y A000045
A000032(n)=F(n+1)+F(n-1).Cf.A060441.%Y A000045 a(k)=A090888(0,k+1)
=A118654(0,k+1)=A118654(1,k-1)=A109754(0,k)=A109754(1,k-1),for k>0.
%Y A000045 Cf.A059929.%Y A000045 A144152[From Gary W.Adamson (qntmpkt
(AT)yahoo.com),Sep 12 2008]
%Y A000045 Sequence in context:A107358 A132636 A039834 this_sequence
A020695 A132916 A069041
%Y A000045 Adjacent sequences:A000042 A000043 A000044 this_sequence
A000046 A000047 A000048
%K A000045 core,nonn,easy,nice,new
%O A000045 0,4
%A A000045 njas

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