A122120 a(n) = 4*a(n-1) + 9*a(n-2), for n>1, with a(0)=1, a(1)=3.

1, 3, 21, 111, 633, 3531, 19821, 111063, 622641, 3490131, 19564293, 109668351, 614752041, 3446023323, 19316861661, 108281656551, 606978381153, 3402448433571, 19072599164661, 106912432560783, 599303122725081

Offset: 0

Author

Philippe Deléham, Oct 19 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,9).

Crossrefs

First differences of A015533.

Binomial transform of A091914.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-4*x-9*x^2) )); // G. C. Greubel, Feb 26 2019

Mathematica

CoefficientList[Series[(1-x)/(1-4*x-9*x^2), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)
nxt[{a_,b_}]:={b,4b+9a}; NestList[nxt,{1,3},20][[All,1]] (* or *) LinearRecurrence[{4,9},{1,3},30] (* Harvey P. Dale, Oct 06 2020 *)

PARI

my(x='x+O('x^30)); Vec((1-x)/(1-4*x-9*x^2)) \\ G. C. Greubel, Feb 26 2019

Sage

((1-x)/(1-4*x-9*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Formula

a(n) = Sum_{k=0..n} 3^(n-k)*A055380(n,k).

G.f.: (1-x)/(1-4*x-9*x^2).

Limit_{n -> oo} a(n+1)/a(n) = 2 + sqrt(13).