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

Author

Stephen Tucker, Jun 17 2018

Keywords

• base
• nonn

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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

PARI

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

Extensions

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