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A122120
a(n) = 4*a(n-1) + 9*a(n-2), for n>1, with a(0)=1, a(1)=3.
1
1, 3, 21, 111, 633, 3531, 19821, 111063, 622641, 3490131, 19564293, 109668351, 614752041, 3446023323, 19316861661, 108281656551, 606978381153, 3402448433571, 19072599164661, 106912432560783, 599303122725081
OFFSET
0,2
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).
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 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-4*x-9*x^2)) \\ G. C. Greubel, Feb 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (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
CROSSREFS
First differences of A015533.
Binomial transform of A091914.
Sequence in context: A054147 A233582 A043012 * A080952 A183404 A309670
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Oct 19 2006
STATUS
approved