A015525 Expansion of x/(1-3*x-8*x^2).

0, 1, 3, 17, 75, 361, 1683, 7937, 37275, 175321, 824163, 3875057, 18218475, 85655881, 402715443, 1893393377, 8901903675, 41852858041, 196773803523, 925144274897, 4349623252875, 20450023957801, 96147057896403

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,8).

Formula

a(n) = 3*a(n-1) + 8*a(n-2).

a(n) = -16^n*(A^n-B^n)/sqrt(41) where A = -1/(3+sqrt(41)) and B = 1/(sqrt(41)-3). - R. J. Mathar, Apr 29 2008

a(n) = -(-8)^n * a(-n) for all n in Z. - Michael Somos, Mar 05 2020

Example

G.f. = x + 3*x^2 + 17*x^3 + 75*x^4 + 361*x^5 + 1683*x^6 + 7937*x^7 + ... - Michael Somos, Mar 05 2020

Mathematica

a[n_]:=(MatrixPower[{{1, 4}, {1, -4}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 12 2012 *)
a[ n_] := With[{m=n-1, t=Sqrt[-8]}, t^m ChebyshevU[m, -t 3/16]]; (* Michael Somos, Mar 05 2020 *)

Prog

(Sage) [lucas_number1(n, 3, -8) for n in range(0, 23)]# Zerinvary Lajos, Apr 22 2009
(Magma) [n le 2 select n-1 else 3*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012
(PARI) x='x+O('x^30); concat([0], Vec(x/(1-3*x-8*x^2))) \\ G. C. Greubel, Jan 01 2017
(PARI) {a(n) = if( n<0, -(-8)^n * a(-n), polcoeff( x / (1 - 3*x - 8*x^2) + x * O(x^n), n))}; /* Michael Somos, Mar 05 2020 */

Crossrefs

Sequence in context: A171875 A356390 A100233 * A062224 A217958 A093568

Adjacent sequences: A015522 A015523 A015524 * A015526 A015527 A015528

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved