A120612 For n>1, a(n) = 2*a(n-1) + 15*a(n-2); a(0)=1, a(1)=1.

1, 1, 17, 49, 353, 1441, 8177, 37969, 198593, 966721, 4912337, 24325489, 122336033, 609554401, 3054149297, 15251614609, 76315468673, 381405156481, 1907542343057, 9536162033329, 47685459212513, 238413348924961, 1192108586037617, 5960417405949649

Offset

0,3

Comments

Characteristic polynomial of matrix M = x^2 - 2x - 15. a(n)/a(n-1) tends to 5, largest eigenvalue of M and a root of the characteristic polynomial.

Binomial transform of [1, 0, 16, 0, 256, 0, 4096, 0, 65536, 0, ...]=: powers of 16 (A001025) with interpolated zeros. - Philippe Deléham, Dec 02 2008

a(n) is the number of compositions of n when there are 1 type of 1 and 16 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (2,15).

Formula

Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term.

From Alexander Adamchuk, Aug 31 2006: (Start)

a(n) = ( 5^n + (-1)^n * 3^n ) / 2.

a(2n+1) = A005059(2n+1).

a(2n) = A081186(2n). (End)

a(n) = Sum_{k=0..n} A098158(n,k)*16^(n-k). - Philippe Deléham, Dec 26 2007

If p(1)=1, and p(i)=16, (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

Example

a(4) = 353 = 2*49 + 15*17 = 2*a(3) + 15*a(2).

Mathematica

Table[(5^n+(-1)^n*3^n)/2, {n, 1, 30}] (* Alexander Adamchuk, Aug 31 2006 *)
a[n_] := (5^n + (-3)^n)/2; Array[a, 24, 0] (* Or *)
CoefficientList[Series[(1 + 15 x)/(1 - 2 x - 15 x^2), {x, 0, 23}], x] (* Or *)
LinearRecurrence[{2, 15}, {1, 1}, 25] (* Or *)
Table[ MatrixPower[{{1, 2}, {8, 1}}, n][[1, 1]], {n, 0, 30}]  (* Robert G. Wilson v, Sep 18 2013 *)

Prog

(PARI) a(n)=([1, 4; 4, 1]^n)[1, 1] \\ Charles R Greathouse IV, Oct 16 2013
(PARI) concat(1, Vec((15*x+1)/(-15*x^2-2*x+1) + O(x^100))) \\ Colin Barker, Mar 12 2014
(PARI) a(n) = ( 5^n + (-1)^n * 3^n ) / 2 \\ Charles R Greathouse IV, May 18 2015

Crossrefs

Cf. A005059, A081186, A059841.

Sequence in context: A146831 A146698 A146706 * A146461 A098329 A160076

Adjacent sequences: A120609 A120610 A120611 * A120613 A120614 A120615

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 17 2006

Extensions

More terms from Alexander Adamchuk, Aug 31 2006

Entry revised by Philippe Deléham, Dec 02 2008

More terms from Colin Barker, Mar 12 2014

Status

approved