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A180666 Golden Triangle sums: a(n)=a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.

%I A180666 #13 Oct 14 2024 05:52:19
%S A180666 0,1,2,6,15,41,106,279,729,1911,5001,13095,34281,89752,234971,615165,
%T A180666 1610520,4216400,11038675,28899630,75660210,198081006,518582802,
%U A180666 1357667406,3554419410,9305590831,24362353076,63781468404
%N A180666 Golden Triangle sums: a(n)=a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.
%C A180666 The a(n) are the Gi2 sums of the Golden Triangle A180662. See A180662 for information about these giraffe and other chess sums.
%H A180666 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1,1,-2,-2,1).
%F A180666 a(n) = a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.
%F A180666 G.f.: (-x)/((x^2-3*x+1)*(x-1)*(x+1)^2*(x^2+1)).
%F A180666 a(n) = Sum_{k=0..floor(n/4)} A180662(n-3*k,n-4*k).
%F A180666 120*a(n) = 8*A001519(n) -10*A087960(n) -9*(-1)^n -15 -6*(n+1)*(-1)^n. - _R. J. Mathar_, Aug 18 2016
%p A180666 nmax:=27: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n)*fibonacci(n+1) od: a(0):=0: a(1):=1: a(2):=2: a(3):=6: for n from 4 to nmax do a(n):=a(n-4)+A001654(n) od: seq(a(n),n=0..nmax);
%p A180666 A180666 := proc(n)
%p A180666     option remember;
%p A180666     if n <=3 then
%p A180666         op(n+1,[0,1,2,6]) ;
%p A180666     else
%p A180666         procname(n-4)+A001654(n) ;
%p A180666     end if;
%p A180666 end proc:
%p A180666 seq(A180666(n),n=0..100 ) ; # _R. J. Mathar_, Aug 18 2016
%t A180666 Take[Total@{#, PadLeft[Drop[#, -4], Length@ #]}, Length@ # - 4] &@ Table[Times @@ Fibonacci@ {n, n + 1}, {n, 0, 31}] (* or *)
%t A180666 CoefficientList[Series[(-x)/((x^2 - 3 x + 1) (x - 1) (x + 1)^2 (x^2 + 1)), {x, 0, 27}], x] (* _Michael De Vlieger_, Aug 18 2016 *)
%Y A180666 Cf. A064831, A180664, A180665, A115730, A180666.
%K A180666 easy,nonn
%O A180666 0,3
%A A180666 _Johannes W. Meijer_, Sep 21 2010