A180665 Golden Triangle sums: a(n)=a(n-2)+A001654(n) with a(0)=0 and a(1)=1.

0, 1, 2, 7, 17, 47, 121, 320, 835, 2190, 5730, 15006, 39282, 102847, 269252, 704917, 1845491, 4831565, 12649195, 33116030, 86698885, 226980636, 594243012, 1555748412, 4073002212, 10663258237, 27916772486, 73087059235

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

The a(n) are the Kn21, Kn22, Kn23, Fi2, and Ze2 sums of the Golden Triangle A180662. Furthermore the a(2*n) are the Kn3, Fi1 (terms doubled) and Ze3 (terms tripled) sums. See A180662 for information about these and other chess sums.

Links

• Index entries for linear recurrences with constant coefficients, signature (2, 3, -3, -2, 1).

Crossrefs

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

Programs

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: for n from 2 to nmax do a(n) := a(n-2) + A001654(n) od: seq(a(n),n=0..nmax);

Formula

a(n) = a(n-2)+A001654(n) with a(0)=0 and a(1)=1.

GF(x) = (-x)/((x-1)*(x+1)^2*(x^2-3*x+1)).

a(n) = ((-1)^(-n-1)*(15+10*n)-25+(16-4*A)*A^(-n-1)+(16-4*B)*B^(-n-1))/100 with A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2.