Re: 1/89 and the Fibonacci sequence.

Sorry about the multiple posts but this was
a bitch to edit!

The mystery of 1/89 and the Fibonacci sequence


1/89 =
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
01123595505617977528089887640449438202247191/
...
With a continuing decimal expansion that has
a period of 44.

Then adding the Fibonacci sequence in this manner --
***OOPs- fixed below***
0112358
>>>>+13
>>>>+>21
>>>>+>>34
>>>>+>>>55
>>>>+>>>>89
>>>>+>>>>144
>>>>+>>>>>233 Which creates a right
>>>>+>>>>>>377 one step offset.
>>>>+>>>>>>>610
>>>>+>>>>>>>>987
>>>>+>>>>>>>>1597
>>>>+>>>>>>>>>2584
>>>>+>>>>>>>>>>4181
>>>>+>>>>>>>>>>>6765
>>>>+>>>>>>>>>>>10946
>>>>+>>>>>>>>>>>>..... etc.
----------------------------------
01123595505617977528089887640449438202247191...
= 1/89?

Will this continue repeating the period of 1/89
no matter how many fibonacci numbers are added
in this manner?

If it does, can it be proved?

Others like 1/80 can be represented as --
The sequence of -- 0,1,2,4,8,16,32,64,128...
>.01248
>+>>16
>+>>>32
>+>>>>64
>+>>>>128
>+>>>>>256
>+>>>>>>512
>+>>>>>>1024
>+>>>>>>>2048
>+>>>>>>>>4096
>+>>>>>>>>>8192
>+>>>>>>>>>16384
>+>>>>>>>>>>32768
>+>>>>>>>>>>>65536
>+>>>>>>>>>>>>...... etc.
----------------------------------
=.0124999999999999999999999999999...
or .0125 = 1/80

Another is the silver mean sequence where the difference
between terms converges on sqrt(2)+1. Found in (OEIS) A000129.

0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461...

Then 1/79 represents the adding together below.

0125
+>>12
+>>>29
+>>>>70
+>>>>169
+>>>>>408
+>>>>>>985
+>>>>>>2378
+>>>>>>>5741
+>>>>>>>13860
+>>>>>>>>33461
+>>>>>>>>>80782
+>>>>>>>>>195025
+>>>>>>>>>>470832
+>>>>>>>>>>1136689
+>>>>>>>>>>>2744210
+>>>>>>>>>>>>6625109
+>>>>>>>>>>>>15994428
+>>>>>>>>>>>>>........ etc.
---------------------------------------------------
0126582278481012658227848101265822784810126582278481...
= 1/79 having a period of 13.

1/69 is this sequence added in the above manner.
0,1,3,10,33,109,360,1189,3927,12970,42837,141481,467280...
In OEIS as A006190.

1/59 also in OEIS as sequence A0010076

If all this is true, what is the sequence to be added
in the above manner for ---
1/78,1/81,1/82,.. etc.?


Dan
.

Relevant Pages

  • 1/89 and the Fibonacci sequence.
    ... Then adding the Fibonacci sequence in this manner -- ... In OEIS as A006190. ...
    (sci.math)
  • Re: 1/89 and the Fibonacci sequence.
    ... Then adding the Fibonacci sequence in this manner -- ... In OEIS as A006190. ...
    (sci.math)
  • Re: iterative-version for computing a Fibonacci number
    ... if you keep multiplying A by itself you get the following sequence: ... You will notice that the sequence of first numbers from each matrix is ... the Fibonacci sequence. ... This technique is used in the Python code here: ...
    (comp.lang.lisp)
  • Re: Bizarritudes
    ... >sequences of digits be as they are. ... Binet's "closed form" for the Fibonacci sequence is ...
    (sci.math)
  • Re: problem with a necklace sequence
    ... context was completely unrelated to the one that appears in the OEIS. ... "Number of aperiodic bracelets (or necklaces) with n red or blue beads ... indeed get the same numbers as in the OEIS sequence. ...
    (sci.math)