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
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