1,1

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.

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

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

Sequence in context: A210426 A231242 A221730 * A160150 A210119 A272186

Adjacent sequences: A306033 A306034 A306035 * A306037 A306038 A306039

base,nonn

Stephen Tucker, Jun 17 2018

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

approved