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

0,2

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A154144 A255942 A355551 * A294959 A290926 A018042

Adjacent sequences: A180661 A180662 A180663 * A180665 A180666 A180667

Keyword

easy,nonn

Author

Johannes W. Meijer, Sep 21 2010

Status

approved