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

0,4

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.

Links

Table of n, a(n) for n=0..79.

Index entries for Colombian or self numbers and related sequences

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{1<k<n, Fib(n-k+1)*floor(a(k)/11)}, where Fib(n)=A000045(n).

Example

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.

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 *)

Crossrefs

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

Sequence in context: A309782 A300999 A074867 * A267809 A010077 A364120

Adjacent sequences: A131294 A131295 A131296 * A131298 A131299 A131300

Keyword

nonn,base

Author

Hieronymus Fischer, Jun 27 2007

Extensions

Incorrect comment removed by Michel Marcus, Apr 29 2018

Status

approved