A206012 - OEIS

%I #39 Sep 08 2022 08:46:01

%S 1,1,1,1,3,2,3,5,8,16,13,21,34,50,105,84,134,218,323,675,541,864,1405,

%T 2080,4349,3485,5565,9050,13399,28014,22449,35848,58297,86311,180456,

%U 144608,230919,375527,555983,1162429,931510,1487493,2419003,3581432,7487928

%N Modular recursion: a(0)=a(1)=a(2)=a(3)=1, thereafter: a(n) equals a(n - 2) + a(n - 3) when n = 0 mod 5, a(n - 1) + a(n - 3) when n = 1 mod 5, a(n - 1) + a(n - 2) when n = 2 mod 5, a(n - 1) + a(n - 4) when n = 3 mod 5, and a(n - 1) + a(n - 2) + a(n - 3) otherwise.

%C This sequence was inspired by the work of _Paul Curtz_ on three part sequences. I did a three part version of this that gave a generating polynomial and got even more variance by adding two more modulo sequences.

%H Vincenzo Librandi, <a href="/A206012/b206012.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,6,0,0,0,0,3,0,0,0,0,-1).

%F G.f.: (x^15 - x^13 + x^12 - 2x^10 - 2x^9 + 2x^8 - x^7 - 3x^6 - 4x^5 + 3x^4 + x^3 + x^2 + x + 1) / (x^15 - 3x^10 - 6x^5 + 1). - _Alois P. Heinz_, Mar 19 2012

%t a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1;a[n_Integer] := a[n]=If[Mod[n, 5] == 0, a[n - 2] + a[n - 3], If[Mod[n, 5] == 1, a[n - 1] + a[n - 3], If[Mod[n, 5] == 2, a[n - 1] + a[n - 2], If[Mod[n, 5] == 3, a[n - 1] + a[n - 4], a[n - 1] + a[n - 2] + a[n - 3]]]]];b = Table[a[n], {n, 0, 50}];(* FindSequenceFunction gives*);Table[c[n] = b[[n]], {n, 1, 16}];c[n_Integer] := c[n] = -c[-15 + n] + c[-10 + n] + 6 c[-5 + n];d = Table[c[n], {n, 1, Length[b]}]

%t CoefficientList[Series[(x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1),{x,0,1001}],x] (* _Vincenzo Librandi_, Apr 01 2012 *)

%o (PARI) Vec((x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 19 2012

%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1))); // _Bruno Berselli_, Mar 20 2012

%Y Cf. A194880, A195240, A185138, A206568, A110897, A110898.

%K nonn,easy,less

%O 0,5

%A _Roger L. Bagula_, Mar 19 2012