A133345 a(n) = 2*a(n-1) + 14*a(n-2) for n>1, a(0)=1, a(1)=1.

1, 1, 16, 46, 316, 1276, 6976, 31816, 161296, 768016, 3794176, 18340576, 89799616, 436367296, 2129929216, 10369000576, 50557010176, 246280028416, 1200358199296, 5848636796416, 28502288382976, 138885491915776

Offset: 0

Author

Philippe Deléham, Dec 21 2007

Keywords

• nonn
• easy

Comments

Binomial transform of A001024 (powers of 15), with interpolated zeros.

a(n) is the number of compositions of n when there are 1 type of 1 and 15 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 (2,14).

Crossrefs

Cf. A001024, A098158.

Programs

Magma

[n le 2 select 1 else 2*(Self(n-1) +7*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

Mathematica

LinearRecurrence[{2,14},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

PARI

Vec((1-x)/(1-2*x-14*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

SageMath

A133345=BinaryRecurrenceSequence(2,14,1,1)
[A133345(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

Formula

G.f.: (1-x)/(1-2*x-14*x^2).

a(n) = Sum_{k=0..n} A098158(n,k)*15^(n-k). - Philippe Deléham, Dec 26 2007

If p[1]=1, and p[i]=15, (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(14). - G. C. Greubel, Oct 15 2022