A083101 a(n) = 2*a(n-1) + 10*a(n-2), with a(0) = 1, a(1) = 12.

1, 12, 34, 188, 716, 3312, 13784, 60688, 259216, 1125312, 4842784, 20938688, 90305216, 389997312, 1683046784, 7266066688, 31362601216, 135385869312, 584397750784, 2522654194688, 10889285897216, 47005113741312, 202903086454784, 875857310322688, 3780745485193216

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

Keywords

• easy
• nonn

Comments

a(n)/A083102(n) converges to sqrt(11).

a(n-1) is the number of compositions of n when there are 1 type of 1 and 11 types of other natural numbers. - Milan Janjic, Aug 13 2010

A133294 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. A083102, A133294.

Programs

Magma

I:=[1,12]; [n le 2 select I[n] else 2*Self(n-1) + 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

Mathematica

CoefficientList[Series[(1+10x)/(1-2x-10x^2), {x, 0, 25}], x]
LinearRecurrence[{2, 10},{1, 12},22] (* Ray Chandler, Sep 23 2015 *)

PARI

x='x+O('x^30); Vec((1+10*x)/(1-2*x-10*x^2)) \\ G. C. Greubel, Jan 08 2018

Formula

a(n+1) = a(n) + 11*A083102(n).

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

If p[1]=1, and p[i]=11, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det A. - Milan Janjic, Apr 29 2010