Re: 1/89 and the Fibonacci sequence-

Not necessary to stay silent about the 1/89 mystery and related mysteries:

Try and explore power series developments like...

1/(1 - x - x^2) = 1 + x + 2x^2 + 3x^3 + 5x^4 + ...,
which gives 1/0.89 = 1.1235... and 1/0.9899 = 1.0102030509....

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
1 = (1 - x - x^2)(1 + x + 2x^2 + 3x^3 + 5x^4 + ...)

1/(1 - x) = 1 + x + x^2 + x^3 + ...,
which gives 1/0.98 = 1.020408163265.... and 14/0.98 = 100/7 = 14.285714....;
1/0.998 = 1.002004008016032064128256513...; 1/0.997 = 1.003009027083...

All power series developments of reciprocals of polynomials yield recurrent relations for the coefficients.

Cheers - Johan E. Mebius

Dan wrote:

>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
>

.

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