A168579 G.f.: 1/(1-x-16*x^2).

1, 1, 17, 33, 305, 833, 5713, 19041, 110449, 415105, 2182289, 8823969, 43740593, 184924097, 884773585, 3843559137, 17999936497, 79496882689, 367495866641, 1639445989665, 7519379855921, 33750515690561, 154060593385297

Offset: 0

Author

Philippe Deléham, Nov 30 2009

Keywords

• nonn
• easy

Comments

The ratio a(n+1)/a(n) converges to (1+sqrt(65))/2 as n approaches infinity. - Felix P. Muga II, Mar 12 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J. Borowska, L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=1, b=4.
• Index entries for linear recurrences with constant coefficients, signature (1,16).

Programs

Mathematica

Join[{a=1,b=1},Table[c=1*b+16*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/(1 - x - 16 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{1,16},{1,1},30] (* Harvey P. Dale, Aug 14 2014 *)

PARI

a(n)=([0,1; 16,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

Formula

a(0)=a(1)=1, a(n) = a(n-1) + 16*a(n-2) for n>1.

a(n) = (1/sqrt(65))*( ((1+sqrt(65))/2)^(n+1) - ((1-sqrt(65))/2)^(n+1) ), for n >= 0 [Binet representation] - Felix P. Muga II, Mar 12 2014

E.g.f.: (1/sqrt(65))*exp(x/2)*( sinh((sqrt(65)/2)*x) + sqrt(65)*cosh((sqrt(65)/2) *x) ). - G. C. Greubel, Jul 26 2016