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

1, 1, 15, 43, 281, 1121, 5895, 26363, 129361, 601441, 2884575, 13587883, 64675241, 305992961, 1452764055, 6883436603, 32652805921, 154790287681, 734067052335, 3480407844523, 16503687369401, 78252676717601

Offset: 0

Author

Philippe Deléham, Dec 21 2007

Keywords

• nonn
• easy

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. - Milan Janjic, Aug 13 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,13).

Crossrefs

Cf. A001023, A098158.

Programs

Magma

[n le 2 select 1 else 2*Self(n-1) +13*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 15 2022

Mathematica

f[n_]:= Simplify[((1+Sqrt[14])^n + (1-Sqrt[14])^n)/2]; Array[f, 25, 0] (* Or *)
CoefficientList[Series[(1+13x)/(1-2x-13x^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 *)

PARI

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

SageMath

A133343=BinaryRecurrenceSequence(2,13,1,1)
[A133343(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

Formula

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

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

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. - Milan Janjic, Apr 29 2010

a(n) = (b*i)^(n-1)*(b*i*ChebyshevU(n, -i/b) - ChebyshevU(n-1, -i/b)), with b = sqrt(13). - G. C. Greubel, Oct 15 2022