A180664 Golden Triangle sums: a(n) = a(n-1) + A001654(n+1) with a(0)=0.

0, 2, 8, 23, 63, 167, 440, 1154, 3024, 7919, 20735, 54287, 142128, 372098, 974168, 2550407, 6677055, 17480759, 45765224, 119814914, 313679520, 821223647, 2149991423, 5628750623, 14736260448, 38580030722, 101003831720

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

The a(n+1) (terms doubled) are the Kn13 sums of the Golden Triangle A180662. See A180662 for information about these knight and other chess sums.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Crossrefs

Cf. A000032, A000045, A005248, A064831, A115730, A180662, A180664, A180665, A180666.

Programs

Magma

[(1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)): n in [0..40]]; // G. C. Greubel, Jan 21 2022

Maple

nmax:=26: with(combinat): for n from 0 to nmax+1 do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: for n from 1 to nmax do a(n) := a(n-1)+A001654(n+1) od: seq(a(n),n=0..nmax);

Mathematica

Table[Sum[Fibonacci[i+2]*Fibonacci[i+3], {i,0,n-1}], {n,0,40}] (* Rigoberto Florez, Jul 07 2020 *)
LinearRecurrence[{3,0,-3,1},{0,2,8,23},30] (* Harvey P. Dale, Mar 30 2023 *)

Sage

[(1/10)*((-1)^n - 15 + 2*lucas_number2(2*n+4,1,-1)) for n in (0..40)] # G. C. Greubel, Jan 21 2022

Formula

a(n+1) = Sum_{k=0..n} A180662(2*n-k+2, k+2).

a(n) = (-15 + (-1)^n + (6-2*A)*A^(-n-1) + (6-2*B)*B^(-n-1))/10 with A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2.

G.f.: (2*x+2*x^2-x^3)/(1-3*x-x^4+3*x^3).

a(n) = Sum_{i=0..n-1} F(i+2)*F(i+3), where F(i) = A000045(i). - Rigoberto Florez, Jul 07 2020

a(n) = (1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)). - G. C. Greubel, Jan 21 2022