A133294 a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.

1, 1, 12, 34, 188, 716, 3312, 13784, 60688, 259216, 1125312, 4842784, 20938688, 90305216, 389997312, 1683046784, 7266066688, 31362601216, 135385869312, 584397750784, 2522654194688, 10889285897216, 47005113741312

Offset: 0

Author

Philippe Deléham, Dec 20 2007

Keywords

• nonn
• easy

Comments

Binomial transform of [1, 0, 11, 0, 121, 0, 1331, 0, 14641, 0, ...]=: powers of 11 (A001020) with interpolated zeros. - Philippe Deléham, Dec 02 2008

A083101 is an essentially identical sequence (with a different start). - N. J. A. Sloane, Dec 31 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,10).

Crossrefs

Cf. A083101, A090042.

Programs

GAP

a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+10*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019

Magma

I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019

Mathematica

a[n_]:= Simplify[((1+Sqrt[11])^n + (1-Sqrt[11])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-10x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 10}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

PARI

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-10*x^2)) \\ G. C. Greubel, Aug 02 2019

Sage

((1-x)/(1-2*x-10*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

a(n) = Sum_{k=0..n} A098158(n,k)*11^(n-k).

G.f.: (1-x)/(1-2*x-10*x^2).

a(n) = A083101(n-1) for n >= 1.

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

Extensions

Terms a(23) onward added by G. C. Greubel, Aug 02 2019