The On-Line Encyclopedia of Integer Sequences (OEIS)

A180666 revision #13

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

(list; graph; refs; listen; history; text; internal format)

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 *)