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.

0, 1, 2, 6, 15, 41, 106, 279, 729, 1911, 5001, 13095, 34281, 89752, 234971, 615165, 1610520, 4216400, 11038675, 28899630, 75660210, 198081006, 518582802, 1357667406, 3554419410, 9305590831, 24362353076, 63781468404

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..27.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1,1,-2,-2,1).

Formula

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

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

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

120*a(n) = 8*A001519(n) -10*A087960(n) -9*(-1)^n -15 -6*(n+1)*(-1)^n. - R. J. Mathar, Aug 18 2016

Maple

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);
A180666 := proc(n)
    option remember;
    if n <=3 then
        op(n+1, [0, 1, 2, 6]) ;
    else
        procname(n-4)+A001654(n) ;
    end if;
end proc:
seq(A180666(n), n=0..100 ) ; # R. J. Mathar, Aug 18 2016

Mathematica

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

Crossrefs

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

Sequence in context: A004664 A074446 A303551 * A280788 A121328 A348012

Adjacent sequences: A180663 A180664 A180665 * A180667 A180668 A180669

Keyword

easy,nonn

Author

Johannes W. Meijer, Sep 21 2010

Status

approved