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A133343 a(n)=2a(n-1)+13a(n-2) for n>1, a(0)=1, a(1)=1 . 6

%I

%S 1,1,15,43,281,1121,5895,26363,129361,601441,2884575,13587883,

%T 64675241,305992961,1452764055,6883436603,32652805921,154790287681,

%U 734067052335,3480407844523,16503687369401,78252676717601

%N a(n)=2a(n-1)+13a(n-2) for n>1, a(0)=1, a(1)=1 .

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

%C 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]

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (2,13).

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

%F a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*14^(n-k). - _Philippe Deléham_, Dec 26 2007

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

%F 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]

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

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

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

%t Table[ MatrixPower[{{1, 2}, {7, 1}}, n][[1, 1]], {n, 0, 30}] (* _Robert G. Wilson v_, Sep 18 2013 *)

%o (PARI) Vec((1-x)/(1-2*x-13*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jan 12 2012

%K nonn,easy

%O 0,3

%A _Philippe Deléham_, Dec 21 2007

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Last modified November 15 06:27 EST 2019. Contains 329144 sequences. (Running on oeis4.)