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

%I A120612 #43 May 31 2022 06:48:06
%S A120612 1,1,17,49,353,1441,8177,37969,198593,966721,4912337,24325489,
%T A120612 122336033,609554401,3054149297,15251614609,76315468673,381405156481,
%U A120612 1907542343057,9536162033329,47685459212513,238413348924961,1192108586037617,5960417405949649
%N A120612 For n>1, a(n) = 2*a(n-1) + 15*a(n-2); a(0)=1, a(1)=1.
%C A120612 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.
%C A120612 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
%C A120612 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
%H A120612 Vincenzo Librandi, <a href="/A120612/b120612.txt">Table of n, a(n) for n = 0..200</a>
%H A120612 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,15).
%F A120612 Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term.
%F A120612 From _Alexander Adamchuk_, Aug 31 2006: (Start)
%F A120612 a(n) = ( 5^n + (-1)^n * 3^n ) / 2.
%F A120612 a(2n+1) = A005059(2n+1).
%F A120612 a(2n) = A081186(2n). (End)
%F A120612 a(n) = Sum_{k=0..n} A098158(n,k)*16^(n-k). - _Philippe Deléham_, Dec 26 2007
%F A120612 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
%e A120612 a(4) = 353 = 2*49 + 15*17 = 2*a(3) + 15*a(2).
%t A120612 Table[(5^n+(-1)^n*3^n)/2,{n,1,30}] (* _Alexander Adamchuk_, Aug 31 2006 *)
%t A120612 a[n_] := (5^n + (-3)^n)/2; Array[a, 24, 0] (* Or *)
%t A120612 CoefficientList[Series[(1 + 15 x)/(1 - 2 x - 15 x^2), {x, 0, 23}], x] (* Or *)
%t A120612 LinearRecurrence[{2, 15}, {1, 1}, 25] (* Or *)
%t A120612 Table[ MatrixPower[{{1, 2}, {8, 1}}, n][[1, 1]], {n, 0, 30}]  (* _Robert G. Wilson v_, Sep 18 2013 *)
%o A120612 (PARI) a(n)=([1,4; 4,1]^n)[1,1] \\ _Charles R Greathouse IV_, Oct 16 2013
%o A120612 (PARI) concat(1, Vec((15*x+1)/(-15*x^2-2*x+1) + O(x^100))) \\ _Colin Barker_, Mar 12 2014
%o A120612 (PARI) a(n) = ( 5^n + (-1)^n * 3^n ) / 2 \\ _Charles R Greathouse IV_, May 18 2015
%Y A120612 Cf. A005059, A081186, A059841.
%K A120612 nonn,easy
%O A120612 0,3
%A A120612 _Gary W. Adamson_, Jun 17 2006
%E A120612 More terms from _Alexander Adamchuk_, Aug 31 2006
%E A120612 Entry revised by _Philippe Deléham_, Dec 02 2008
%E A120612 More terms from _Colin Barker_, Mar 12 2014