A131297 a(n) = ds_11(a(n-1))+ds_11(a(n-2)), a(0)=0, a(1)=1; where ds_11=digital sum base 11.

0, 1, 1, 2, 3, 5, 8, 13, 11, 4, 5, 9, 14, 13, 7, 10, 17, 17, 14, 11, 5, 6, 11, 7, 8, 15, 13, 8, 11, 9, 10, 19, 19, 18, 17, 15, 12, 7, 9, 16, 15, 11, 6, 7, 13, 10, 13, 13, 6, 9, 15, 14, 9, 13, 12, 5, 7, 12, 9, 11, 10, 11, 11, 2, 3, 5, 8, 13, 11, 4, 5, 9, 14, 13, 7, 10, 17, 17, 14, 11

Offset: 0

Author

Hieronymus Fischer, Jun 27 2007

Keywords

• nonn
• base

Comments

The digital sum analog (in base 11) of the Fibonacci recurrence.

When starting from index n=3, periodic with Pisano period A001175(10)=60.

a(n) and Fib(n)=A000045(n) are congruent modulo 10 which implies that (a(n) mod 10) is equal to (Fib(n) mod 10)=A003893(n). Thus (a(n) mod 10) is periodic with the Pisano period A001175(10)=60 too.

a(n)==A074867(n) modulo 10 (A074867(n)=digital product analog base 10 of the Fibonacci recurrence).

For general bases p>2, we have the inequality 2<=a(n)<=2p-3 (for n>2). Actually, a(n)<=19=A131319(11) for the base p=11.

Examples

			a(10)=5, since a(8)=11=10(base 11), ds_11(11)=1,
a(9)=4, ds_11(4)=4 and so a(10)=1+4.
		

Links

• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A131297 A000045, A010073, A010074, A010075, A010076, A010077, A131294, A131295, A131296, A131318, A131319, A131320.

Programs

Mathematica

nxt[{a_,b_}]:={b,Total[IntegerDigits[a,11]]+Total[IntegerDigits[b,11]]}; NestList[nxt,{0,1},80][[All,1]] (* or *) PadRight[{0,1,1},80,{10,11,11,2,3,5,8,13,11,4,5,9,14,13,7,10,17,17,14,11,5,6,11,7,8,15,13,8,11,9,10,19,19,18,17,15,12,7,9,16,15,11,6,7,13,10,13,13,6,9,15,14,9,13,12,5,7,12,9,11}] (* Harvey P. Dale, Jul 24 2017 *)

Formula

a(n) = a(n-1)+a(n-2)-10*(floor(a(n-1)/11)+floor(a(n-2)/11)).

a(n) = floor(a(n-1)/11)+floor(a(n-2)/11)+(a(n-1)mod 11)+(a(n-2)mod 11).

a(n) = Fib(n)-10*sum{1A000045(n).

Extensions

Incorrect comment removed by Michel Marcus, Apr 29 2018