A021093 -id:A021093 - cp's OEIS frontend

A306036 a(n) is the period of the decimal expansion of 1/(100^n - 10^n - 1).

44, 468, 496620, 16090340, 2499916380, 499999499999, 47368416315780, 71942445323740, 71428571357142857, 2413792682560590100, 661025543433488700, 998035336547180189380, 9826562531691739684620, 3086415088517393302531635, 33093525179856082014388489204

Offset: 1

Stephen Tucker, Jun 17 2018

It appears that when Fibonacci numbers are written in base 10 diagonally (from top left to bottom right) such that each lower number is n digits farther to the right than its neighbor above, and the columns of digits are summed, the resulting total digit string recurs after a(n) digits.

			For n = 1: the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, when written in a diagonal, with each number 1 digit farther to the right than its predecessor, and the columns summed, gives the digit string 112359... . After this is extended to 44 numbers, the digit string has another occurrence of 112359. I conjecture that this is because the reciprocal of 89 has a period of 44 digits. It also demonstrates an amazing property of Fibonacci numbers.

Cf. A007732, A021093 (1/89), A086695.

Array[MultiplicativeOrder[10, (100^# - 10^# - 1)] &, 15] (* Michael De Vlieger, Jun 29 2018 *)

a(n) = znorder(Mod(10, 100^n-10^n-1)) \\ Felix Fröhlich, Jun 18 2018

More terms from Jon E. Schoenfield and Felix Fröhlich, Jun 18 2018

A307991 Fibonacci numbers of the form k^2 - k - 1 with integer k.

Original entry on oeis.org

1, 5, 55, 89

Offset: 1

Amiram Eldar, May 09 2019

The corresponding values of k are 2, 3, 8, 10.
Intersection of A000045 and A028387.
Also Fibonacci numbers whose reciprocals equal to Sum_{i>=1} F(i)/k^(i+1), where F(i) is the i-th Fibonacci number.
de Weger proved that there are no other terms.

			89 is in the sequence since 89 = 10^2 - 10 - 1 or equivalently 1/89 = 1/10^2 + 1/10^3 + 2/10^4 + 3/10^5 + 5/10^6 + ... This is why the first digits of the decimal expansion of 1/89 = 0.011235... are the first terms of the Fibonacci sequence.

Fenton Stancliff, A curious property of a_11, Scripta Math., Vol. 19 (1953), p. 126.

B. M. M. de Weger, A Curious Property of the Eleventh Fibonacci Number, The Rocky Mountain Journal of Mathematics, Vol. 25, No. 3 (1995), pp. 977-994.

Cf. A000045, A021093, A028387, A039595, A165900, A227875.

Select[Fibonacci[Range[2, 20]], IntegerQ[Sqrt[4# + 5]] &]

cp's OEIS Frontend

A306036 a(n) is the period of the decimal expansion of 1/(100^n - 10^n - 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions