Re: Whats with this sequence?

[Danny]
This sequence is the difference of each triangle
number and the corresponding Fibonacci number.
e.g.
Starting with the first triangle number and the first
Fibonacci f(1) number --

t---f = difference
1 - 1 =0
3 - 1 =2
6 - 2 =4
10- 3 =7
15- 5 =10
etc.
Gives this resulting sequence --

0,2,4,7,10,13,15,15,11,0,-23,-66,-142,-272,
-490,-851,-1444,-2413,...
Note: the 10th term (0) which is 55-55 = t(10)-f(10)= 0.

All fine and good but,why on the third delta row
beneath this sequence the Fibonacci sequence appears?

Because it has to? ;-) Here's a conventional table of differences:

T 1 3 6 10 15 21 28 36 45 55 66 78 91
F 1 1 2 3 5 8 13 21 34 55 89 144 233
T-F 0 2 4 7 10 13 15 15 11 0 -23 -66 -142

d1 2 2 3 3 3 2 0 -4 -11 -23 -43 -76
d2 0 1 0 0 -1 -2 -4 -7 -12 -20 -33
d3 1 -1 0 -1 -1 -2 -3 -5 -8 -13
d4 -2 1 -1 0 -1 -1 -2 -3 -5
d5 3 -2 1 -1 0 -1 -1 -2
d6 -5 3 -2 1 -1 0 -1
d7 8 -5 3 -2 1 -1
d8 -13 8 -5 3 -2
d9 21 -13 8 -5
d10 -34 21 -13
d11 55 -34

You can find the Fibonacci sequence all over the place in that, like the
negation of it in the tail end of row d3, and with alternating signs in the
first column starting at row d3; etc.

Except for the leading 1,1 the 3rd delta sequence is
the Fibonacci sequence excluding the double entry of[1,1]
at the beginning the following starts after the [1,1]--
0,1,1,2,3,5,8,13,21,34,55,89,144...

It just seems strange that this sequence would produce
the Fibonacci sequence in one of its deltas!
Also why did it not list the triangle numbers instead
on say the same or some other delta row?

I expect the mystery will go away if you compute similar difference tables
for the triangle numbers /alone/, and again for the negation of the
Fibonacci sequence alone. The difference table for T-F is just the sum of
the difference table for T and the difference table for -F (that has nothing
to do with T and F specifically: show that the difference table for A+B is
the sum of the difference tables for A and for B, for any sequences A and
B).

Note that the difference table for the triangle numbers is entirely zero
from the third row onward:

2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 0
0 0
0

OTOH, the difference table for the negation of the Fibonacci numbers
"reproduces itself" in weird and wonderful ways because of the relation
between the way Fibonacci numbers are computed and the way difference tables
are constructed (one adds consective pairs, the other subtracts them):

0 -1 -1 -2 -3 -5 -8 -13 -21 -34 -55 -89 -144
-1 0 -1 -1 -2 -3 -5 -8 -13 -21 -34 -55
1 -1 0 -1 -1 -2 -3 -5 -8 -13 -21
-2 1 -1 0 -1 -1 -2 -3 -5 -8
3 -2 1 -1 0 -1 -1 -2 -3
-5 3 -2 1 -1 0 -1 -1
8 -5 3 -2 1 -1 0
-13 8 -5 3 -2 1
21 -13 8 -5 3
-34 21 -13 8
55 -34 21
-89 55
144

Note that the T-F difference table is identical to the -F difference table
(just above) from its 3rd row onward -- and it has to be because the T
difference table is all zeroes from its 3rd row onward.

Similarly, if you take T-S for /any/ sequence S, the difference table for
T-S will be identical to the difference table for S alone from the 3rd row
onward. The only special thing about the triangle numbers here is that its
differences eventually all become 0, and that's true of any sequence S where
S_i can be expressed as a polynomial in i (in general, if it's a polynomial
of degree N, all entries are 0 from difference row N+1 onward; since T_i can
be expressed by a quadratic in i, its differences became 0 starting at
difference row 2+1 = 3).


.

Relevant Pages

  • Re: c language
    ... this sequence was described by the Indian mathematicians Gopala and Hemachandra in 1150, who were investigating the possible ways of exactly bin packing items of length 1 and 2. ... In the West, it was first studied by Leonardo of Pisa, who was also known as Fibonacci, to describe the growth of an idealised rabbit population. ... Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. ... For example, the solutions to reaction-diffusion differential equations can show such a patterning; in biology, genes often express themselves through gene regulatory networks, that is, in terms of several enzymes controlling a reaction, which can be modelled with reaction-diffusion equations. ...
    (comp.programming)
  • Re: Whats with this sequence?
    ... number and the corresponding Fibonacci number. ... Starting with the first triangle number and the first ... Gives this resulting sequence -- ... All fine and good but,why on the third delta row ...
    (sci.math)
  • ++var VS var++
    ... that ++var is faster than var++ as it just adds one and returns the value, ... The above code generates 17 seconds if the compiler is using optimising and ... As I previously said, I was experimenting with the fibonacci sequence, so I ...
    (alt.comp.lang.learn.c-cpp)
  • Re: 1/89 and the Fibonacci sequence-
    ... You will see the Fibonacci numbers being generated when you develop the power series, and you will see them to fall into place when you multiply out the identity ... All power series developments of reciprocals of polynomials yield recurrent relations for the coefficients. ... >>Then adding the Fibonacci sequence in this manner -- ... >>Will this continue repeating the period of 1/89>no matter how many fibonacci numbers are added>in this manner? ...
    (sci.math)
  • Re: Fibonacci program
    ... spartacus wrote: ... i have to program a cobol program where the prompt lists the fibonacci ... number sequence with a maximum of 9 digits. ... So you are a newbie at school and your prof/teacher throws a zinger at you and you really don't know which way is up - 'Fibonacci', 'Compute', 'On Size Error' ..... ...
    (comp.lang.cobol)