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Golden Triangle sums: a(n)=a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.
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%I #13 Oct 14 2024 05:52:19

%S 0,1,2,6,15,41,106,279,729,1911,5001,13095,34281,89752,234971,615165,

%T 1610520,4216400,11038675,28899630,75660210,198081006,518582802,

%U 1357667406,3554419410,9305590831,24362353076,63781468404

%N 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 The a(n) are the Gi2 sums of the Golden Triangle A180662. See A180662 for information about these giraffe and other chess sums.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1,1,-2,-2,1).

%F a(n) = a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.

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

%F a(n) = Sum_{k=0..floor(n/4)} A180662(n-3*k,n-4*k).

%F 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 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 := proc(n)

%p option remember;

%p if n <=3 then

%p op(n+1,[0,1,2,6]) ;

%p else

%p procname(n-4)+A001654(n) ;

%p end if;

%p end proc:

%p seq(A180666(n),n=0..100 ) ; # _R. J. Mathar_, Aug 18 2016

%t Take[Total@{#, PadLeft[Drop[#, -4], Length@ #]}, Length@ # - 4] &@ Table[Times @@ Fibonacci@ {n, n + 1}, {n, 0, 31}] (* or *)

%t 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 Cf. A064831, A180664, A180665, A115730, A180666.

%K easy,nonn

%O 0,3

%A _Johannes W. Meijer_, Sep 21 2010