A105964 - OEIS

%I #17 Mar 08 2024 12:15:23

%S 0,1,0,0,1,0,0,1,1,0,2,0,0,1,2,0,3,0,0,1,3,0,4,0,0,1,4,0,5,0,0,1,5,0,

%T 6,0,0,1,6,0,7,0,0,1,7,0,8,0,0,1,8,0,9,0,0,1,9,0,10,0,0,1,10,0,11,0,0,

%U 1,11,0,12,0,0,1,12,0,13,0,0,1,13,0,14,0,0,1,14,0,15,0,0,1,15,0,16,0,0,1

%N Expansion of x*(1+x^3-x^6+x^7)/(1-x^6)^2.

%C Floretion Algebra Multiplication Program, FAMP Code: 2jbasekfizrokseq[.5'i + .5'ii' - .5'ij' + .5'ik'], RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + 1 (internal program code). FizType: 'i, 'j, 'k.

%H Colin Barker, <a href="/A105964/b105964.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

%F G.f.: x*(1+x^3-x^6+x^7)/((1-x)*(1+x)*(1+x+x^2)*(1-x+x^2))^2.

%F a(n) = 2*a(n-6) - a(n-12) for n>11. - _Colin Barker_, May 13 2019

%p seq(coeff(series(x*(1+x^3-x^6+x^7)/(1-x^6)^2, x, n+1), x, n), n = 0..100); # _G. C. Greubel_, Jan 15 2020

%t CoefficientList[Series[x(1+x^3-x^6+x^7)/(1-x^6)^2,{x,0,100}],x] (* or *) LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{0,1,0,0,1,0,0,1,1,0,2,0},100] (* _Harvey P. Dale_, Aug 08 2019 *)

%o (PARI) concat(0, Vec(x*(1 +x^3 -x^6 +x^7)/(1-x^6)^2 + O(x^100))) \\ _Colin Barker_, May 13 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( x*(1+x^3-x^6+x^7)/(1-x^6)^2 )); // _G. C. Greubel_, Jan 15 2020

%o (SageMath)

%o def A105964_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x*(1+x^3-x^6+x^7)/(1-x^6)^2 ).list()

%o A105964_list(100) # _G. C. Greubel_, Jan 15 2020

%o (GAP) a:=[0,1,0,0,1,0,0,1,1,0,2,0];; for n in [13..100] do a[n]:=2*a[n-6]-a[n-12]; od; a; # _G. C. Greubel_, Jan 15 2020

%K nonn,easy

%O 0,11

%A _Creighton Dement_, Apr 28 2005