A179135 a(n) = (3-sqrt(5))*((3+sqrt(5))/10)^(-n)/2 + (3+sqrt(5))*((3-sqrt(5))/10)^(-n)/2.

3, 35, 450, 5875, 76875, 1006250, 13171875, 172421875, 2257031250, 29544921875, 386748046875, 5062597656250, 66270263671875, 867489013671875, 11355578613281250, 148646453857421875, 1945807342529296875, 25470948791503906250, 333419048309326171875, 4364512004852294921875

Offset

0,1

Links

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (15,-25).

Formula

a(n) = A178381(4*n+3).

G.f.: (3-10*z)/(1-15*z+25*z^2).

Limit_{k->oo} a(n+k)/a(k) = A000351(n)*A130196(n)/(A128052(n) - A167808(2*n)*sqrt(5)).

Limit_{n->oo} A128052(n)/A167808(2*n) = sqrt(5).

a(n) = 5^n*Lucas(2*(n+1)). - Ehren Metcalfe, Apr 22 2018

a(n) = (1/5) * Sum_{k=0..2*n+2} binomial(2*n+2, k) * Lucas(2*k), where Lucas = A000032. - Amiram Eldar, Jan 18 2026

Maple

with(GraphTheory): nmax:=72; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1, k], k=1..P); od: for n from 0 to nmax/4-1 do a(n):= A178381(4*n+3) od: seq(a(n), n=0..nmax/4-1);

Mathematica

LinearRecurrence[{15, -25}, {3, 35}, 20] (* Harvey P. Dale, Jan 10 2026 *)

Crossrefs

Cf. A000032, A000351, A109106, A128052, A130196, A167808, A178381.

Sequence in context: A006767 A065929 A161495 * A100033 A259557 A184554

Adjacent sequences: A179132 A179133 A179134 * A179136 A179137 A179138

Keyword

easy,nonn

Author

Johannes W. Meijer, Jul 01 2010

Status

approved