A175561 Period of the decimal representation of 1/Fibonacci(n).

1, 1, 1, 1, 6, 6, 16, 2, 44, 1, 232, 84, 60, 138, 133, 144, 336, 10, 420, 396, 28656, 66, 1500, 3016, 5616, 84, 514228, 60, 335824, 152214, 9900, 67830, 70980, 7632, 4440, 261744, 27144, 30, 504628, 420, 108373609, 47124, 9120, 3295440, 2971215072, 836418

Offset

3,5

Comments

With regard to A175550, where the period of 1/Fibonacci(n) is concerned with the Fibonacci numbers such that gcd(Fibonacci(n), 10) = 1 only, this sequence is complete.

Links

Table of n, a(n) for n=3..48.

Formula

a(n) = A007732(A000045(n)).

Example

a(8) = 6 because Fibonacci(8) = 21, and 1/21 = 0. 047619 047619 047619 047619 ... => period = 6;
a(9) = 16 because Fibonacci(9) = 34, and 1/34 = 0.0 2941176470588235 2941176470588235 2941176470588235 ... => period = 16.

Mathematica

Table[MultiplicativeOrder[10, n/Times @@ ({2, 5}^IntegerExponent[n, {2, 5}])], {n, Fibonacci[Range[3, 50]]}] (* Amiram Eldar, May 26 2024 *)

Prog

(PARI) a(n) = my(f=fibonacci(n)); znorder(Mod(10, f/2^valuation(f, 2)/5^valuation(f, 5))); \\ Michel Marcus, May 26 2024

Crossrefs

Cf. A000045, A007732, A175550, A175562.

Sequence in context: A290931 A257372 A058563 * A341202 A315810 A315811

Adjacent sequences: A175558 A175559 A175560 * A175562 A175563 A175564

Keyword

nonn,base

Author

Michel Lagneau, Jul 02 2010

Extensions

a(23) onward corrected by Sean A. Irvine, May 26 2024

Status

approved