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A090042 a(n)=2a(n-1)+11a(n-2), a(0)=1, a(1)=1. 5
1, 1, 13, 37, 217, 841, 4069, 17389, 79537, 350353, 1575613, 7005109, 31341961, 139740121, 624241813, 2785624957, 12437909857, 55517694241, 247852396909, 1106399430469, 4939175226937, 22048744189033, 98428415874373 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Binomial transform of A001021 (powers of 12), with interpolated zeros.

a(n), n>0 = term (1,1) in the n-th power of the 2x2 matrix [1,3; 4,1]. [From Gary W. Adamson, Aug 06 2010]

a(n) is the number of compositions of n when there are 1 type of 1 and 12 types of other natural numbers. [From Milan Janjic, Aug 13 2010]

LINKS

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (2,11).

FORMULA

E.g.f.: exp(x)cosh(2sqrt(3)x); a(n)=(1+2sqrt(3))^n/2 + (1-2sqrt(3))^n/2.

a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*12^(n-k). - Philippe Deléham, Dec 26 2007

If p[1]=1, and p[i]=12, (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)=det A. [From Milan Janjic, Apr 29 2010]

MATHEMATICA

a[n_] := Simplify[((1 + Sqrt[11])^n + (1 - Sqrt[11])^n)/2]; Array[a, 25, 0] (* Or *) CoefficientList[Series[(1 + 10 x)/(1 - 2 x - 10 x^2), {x, 0, 23}], x] (* Or *) Table[ MatrixPower[{{1, 2}, {6, 1}}, n][[1, 1]], {n, 0, 25}] (* Robert G. Wilson v, Sep 18 2013 *)

LinearRecurrence[{2, 11}, {1, 1}, 25] (* Ray Chandler, Aug 01 2015 *)

CROSSREFS

Sequence in context: A193646 A155236 A155277 * A266882 A078952 A206279

Adjacent sequences:  A090039 A090040 A090041 * A090043 A090044 A090045

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 20 2003

STATUS

approved

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Last modified December 10 03:42 EST 2016. Contains 278993 sequences.