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A133343 a(n)=2a(n-1)+13a(n-2) for n>1, a(0)=1, a(1)=1 . 6
1, 1, 15, 43, 281, 1121, 5895, 26363, 129361, 601441, 2884575, 13587883, 64675241, 305992961, 1452764055, 6883436603, 32652805921, 154790287681, 734067052335, 3480407844523, 16503687369401, 78252676717601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Binomial transform of A001023 (powers of 14), with interpolated zeros .

a(n) is the number of compositions of n when there are 1 type of 1 and 14 types of other natural numbers. [From Milan Janjic, Aug 13 2010]

LINKS

Table of n, a(n) for n=0..21.

Index entries for linear recurrences with constant coefficients, signature (2,13).

FORMULA

G.f.: (1-x)/(1-2x-13x^2).

a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*14^(n-k). - Philippe Deléham, Dec 26 2007

a(n)=(1/2)*[1-sqrt(14)]^n+(1/2)*[1+sqrt(14)]^n, n>=0 - Paolo P. Lava, Jun 10 2008

If p[1]=1, and p[i]=14, (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

f[n_] := Simplify[((1 + Sqrt[14])^n + (1 - Sqrt[14])^n)/2]; Array[f, 25, 0] (* Or *)

CoefficientList[Series[(1 + 13 x)/(1 - 2 x - 13 x^2), {x, 0, 23}], x] (* Or *)

LinearRecurrence[{2, 13}, {1, 1}, 25] (* Or *)

Table[ MatrixPower[{{1, 2}, {7, 1}}, n][[1, 1]], {n, 0, 30}]  (* Robert G. Wilson v, Sep 18 2013 *)

PROG

(PARI) Vec((1-x)/(1-2*x-13*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

CROSSREFS

Sequence in context: A204734 A126369 A193647 * A027845 A201810 A292018

Adjacent sequences:  A133340 A133341 A133342 * A133344 A133345 A133346

KEYWORD

nonn,easy

AUTHOR

Philippe Deléham, Dec 21 2007

STATUS

approved

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)