A131297 -id:A131297 - cp's OEIS frontend

A010073 a(n) = sum of base-6 digits of a(n-1) + sum of base-6 digits of a(n-2); a(0)=0, a(1)=1.

0, 1, 1, 2, 3, 5, 8, 8, 6, 4, 5, 9, 9, 8, 7, 5, 7, 7, 4, 6, 5, 6, 6, 2, 3, 5, 8, 8, 6, 4, 5, 9, 9, 8, 7, 5, 7, 7, 4, 6, 5, 6, 6, 2, 3, 5, 8, 8, 6, 4, 5, 9, 9, 8, 7, 5, 7, 7, 4, 6, 5, 6, 6, 2, 3, 5, 8, 8, 6, 4, 5, 9, 9, 8, 7, 5, 7, 7, 4, 6, 5, 6, 6

Offset: 0

N. J. A. Sloane, Leonid Broukhis

The digital sum analog (in base 6) of the Fibonacci recurrence. - Hieronymus Fischer, Jun 27 2007
For general bases p > 2, we have the inequality 2 <= a(n) <= 2p-3 (for n > 2). Actually, a(n) <= 9 = A131319(6) for the base p=6. - Hieronymus Fischer, Jun 27 2007
a(n) and Fibonacci(n)=A000045(n) are congruent modulo 5 which implies that (a(n) mod 5) is equal to (Fibonacci(n) mod 5) = A082116(n) (for n > 0). Thus (a(n) mod 6) is periodic with the Pisano period A001175(5)=20. - Hieronymus Fischer, Jun 27 2007

Index entries for Colombian or self numbers and related sequences

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

[0] cat [n le 2 select 1 else Self(n-1)+Self(n-2)-5*((Self(n-1) div 6)+(Self(n-2) div 6)): n in [1..100]]; // Vincenzo Librandi, Jul 11 2015

nxt[{a_,b_,c_}]:={b,c,Total[IntegerDigits[c,6]]+Total[ IntegerDigits[ b,6]]}; Transpose[NestList[nxt,{0,1,1},90]][[1]] (* Harvey P. Dale, Oct 09 2014 *)

lista(nn) = {va = vector(nn); va[2] = 1; for (n=3, nn, va[n] = sumdigits(va[n-1], 6) + sumdigits(va[n-2], 6);); va;} \\ Michel Marcus, Apr 24 2018

Periodic from n=3 with period 20. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = a(n-1) + a(n-2) - 5*(floor(a(n-1)/6) + floor(a(n-2)/6)). - Hieronymus Fischer, Jun 27 2007
a(n) = floor(a(n-1)/6) + floor(a(n-2)/6) + (a(n-1) mod 6) + (a(n-2) mod 6). - Hieronymus Fischer, Jun 27 2007
a(n) = (a(n-1) + a(n-2) + 5*(A010875(a(n-1)) + A010875(a(n-2))))/6. - Hieronymus Fischer, Jun 27 2007
a(n) = Fibonacci(n) - 5*Sum_{k=2..n-1} Fibonacci(n-k+1)*floor(a(k)/6). - Hieronymus Fischer, Jun 27 2007

Incorrect comment removed by Michel Marcus, Apr 28 2018

A074867 a(n) = M(a(n-1)) + M(a(n-2)) where a(1)=a(2)=1 and M(k) is the product of the digits of k in base 10.

Original entry on oeis.org

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

Offset: 1

Felice Russo, Sep 11 2002

Periodic with least period 60. - Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003
From Hieronymus Fischer, Jul 01 2007: (Start)
The digital product analog (in base 10) of the Fibonacci recurrence.
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.
a(n)==A131297(n) modulo 10 (A131297(n)=digital sum analog base 11 of the Fibonacci recurrence).
For general bases p>1, we have the inequality 1<=a(n)<=2p-2 (for n>0). Actually, a(n)<=18.
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

nxt[{a_,b_}]:={b,Times@@IntegerDigits[a]+Times@@IntegerDigits[b]}; Transpose[ NestList[nxt,{1,1},90]][[1]] (* Harvey P. Dale, Feb 01 2015 *)

From Hieronymus Fischer, Jul 01 2007: (Start)
a(n) = a(n-1)+a(n-2)-10*(floor(a(n-1)/10)+floor(a(n-2)/10)). This is valid, since a(n)<100.
a(n) = ds_10(a(n-1))+ds_10(a(n-2))-(floor(a(n-1)/10)+floor(a(n-2)/10)) where ds_10(x) is the digital sum of x in base 10.
a(n) = (a(n-1)mod 10)+(a(n-2)mod 10) = A010879(a(n-1))+A010879(a(n-2)).
a(n) = A131297(n) if A131297(n)<=10.
a(n) = Fib(n)-10*sum{1A000045(n).
a(n) = A000045(n)-10*sum{1A000045(n-k+1)*A059995(a(k))}. (End)

More terms from Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003

Definition adapted to offset by Georg Fischer, Jun 18 2021

cp's OEIS Frontend

A010073 a(n) = sum of base-6 digits of a(n-1) + sum of base-6 digits of a(n-2); a(0)=0, a(1)=1.

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