A131319 Maximal value arising in the sequence S(n) representing the digital sum analog base n of the Fibonacci recurrence.

1, 2, 3, 5, 5, 9, 11, 13, 13, 17, 19, 13, 19, 25, 27, 26, 25, 33, 35, 32, 33, 34, 35, 45, 41, 49, 51, 53, 43, 34, 54, 51, 56, 56, 67, 61, 55, 73, 55, 67, 69, 81, 65, 85, 67, 82, 91, 93, 89, 97, 99, 88, 89, 105, 107, 89, 97, 97, 89, 98, 111, 121, 109, 118, 105, 129, 112

Offset: 1

Author

Hieronymus Fischer, Jul 08 2007

Keywords

• nonn
• base

Comments

The respective period lengths of S(n) are given by A001175(n-1) (which is the Pisano period to n-1) for n>=2.

The inequality a(n)<=2n-3 holds for n>2.

a(n)=2n-3 infinitely often; lim sup a(n)/n=2 for n-->oo.

Examples

			a(3)=3, since the digital sum analog base 3 of the Fibonacci sequence is S(3)=0,1,1,2,3,3,2,3,3,... where the pattern {2,3,3} is the periodic part (see A131294) and so has a maximal value of 3.
a(9)=13 because the pattern base 9 is {2,3,5,8,13,13,10,7,9,8,9,9} (see A010076) where the maximal value is 13.
		

Crossrefs

Cf. A000032, A000045, A131318, A131320.

See A010074, A010075, A010076, A010077, A131294, A131295, A131296, A131297 for the definition of the digital sum analog of the Fibonacci recurrence(in different bases).

Formula

For n=Lucas(2m)=A000032(2m) with m>0, we have a(n)=2n-3.

a(n)=2n-A131320(n).