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%I #48 Sep 08 2022 08:44:37
%S 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,
%T 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,
%U 6,4,5,9,9,8,7,5,7,7,4,6,5,6,6
%N 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.
%C The digital sum analog (in base 6) of the Fibonacci recurrence. - _Hieronymus Fischer_, Jun 27 2007
%C 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
%C 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
%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F Periodic from n=3 with period 20. - _Franklin T. Adams-Watters_, Mar 13 2006
%F a(n) = a(n-1) + a(n-2) - 5*(floor(a(n-1)/6) + floor(a(n-2)/6)). - _Hieronymus Fischer_, Jun 27 2007
%F 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
%F a(n) = (a(n-1) + a(n-2) + 5*(A010875(a(n-1)) + A010875(a(n-2))))/6. - _Hieronymus Fischer_, Jun 27 2007
%F a(n) = Fibonacci(n) - 5*Sum_{k=2..n-1} Fibonacci(n-k+1)*floor(a(k)/6). - _Hieronymus Fischer_, Jun 27 2007
%t 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 *)
%o (Magma) [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
%o (PARI) 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
%Y Cf. A000045, A010074, A010075, A010076, A010077, A131294, A131295, A131296, A131297, A131318, A131319, A131320.
%K nonn,base
%O 0,4
%A _N. J. A. Sloane_, _Leonid Broukhis_
%E Incorrect comment removed by _Michel Marcus_, Apr 28 2018
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