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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..15.

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A210426 A231242 A221730 * A160150 A210119 A272186

Adjacent sequences: A306033 A306034 A306035 * A306037 A306038 A306039

Keyword

base,nonn

Author

Stephen Tucker, Jun 17 2018

Extensions

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

Status

approved