A155144 a(n) = 9*a(n-1) + 9*a(n-2), n>2; a(0)=1, a(1)=8, a(2)=80.

1, 8, 80, 792, 7848, 77760, 770472, 7634088, 75641040, 749476152, 7426054728, 73579777920, 729052493832, 7223690445768, 71574686456400, 709185392119512, 7026840707183208, 69624234893724480, 689859680408169192

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..998

Index entries for linear recurrences with constant coefficients, signature (9,9).

Formula

G.f.: (1-x-x^2)/(1-9*x-9*x^2).

a(n) = 4*( ((9 + 3*sqrt(13))/2)^(n-1) + ((9 - 3*sqrt(13))/2)^(n-1) ) + (44/39)*sqrt(13)*( ((9 + 3*sqrt(13))/2)^(n-1) - ((9 - 3*sqrt(13))/2)^(n-1) ) + (1/9)*[binomial(2*n,n) mod 2], with n>=0. - Paolo P. Lava, Jan 26 2009

From G. C. Greubel, Mar 24 2021: (Start)

a(n) = (1/9)*[n=0] - 8*3^(n-2)*ChebyshevU(n, -3*i/2).

a(n) = (1/9)*[n=0] + 8*3^(n-2)*Fibonacci(n+1, 3). (End)

Maple

m:=30; S:=series( (1-x-x^2)/(1-9*x-9*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 24 2021

Mathematica

Join[{1}, LinearRecurrence[{9, 9}, {8, 80}, 20]] (* or *) CoefficientList[ Series[ (1-x-x^2)/(1-9x-9x^2), {x, 0, 20}], x] (* Harvey P. Dale, Jun 19 2011 *)

Prog

(Magma) [1] cat [n le 2 select 8*(9*n-8) else 9*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 24 2021
(Sage) [1]+[-8*(3*i)^(n-2)*chebyshev_U(n, -3*i/2) for n in (1..30)] # G. C. Greubel, Mar 24 2021

Crossrefs

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), this sequence (m=9), A155157 (m=10).

Sequence in context: A290874 A024101 A291181 * A299871 A136949 A346178

Adjacent sequences: A155141 A155142 A155143 * A155145 A155146 A155147

Keyword

nonn

Author

Philippe Deléham, Jan 21 2009

Status

approved