A085504 Horadam sequence (0,1,9,3).

0, 1, 18, 81, 405, 1944, 9477, 45927, 223074, 1082565, 5255361, 25509168, 123825753, 601059771, 2917611090, 14162371209, 68745613437, 333698181192, 1619805064509, 7862698824255, 38166342053346, 185263315578333, 899287025215113, 4365230915850336

Offset

0,3

Comments

Lim_{n->infinity} a(n)/a(n-1) = (3/2)*(1 + sqrt(5)), which can also be written as phi^2 + 2*phi - 1, phi^3 + phi - 1, phi + sqrt(5) + 1, 3*phi, 3*phi^2 - 3, phi^4 - 2 and lim_{n->infinity} (3/2)*(1 + Lucas(n)/Fibonacci(n)).

Links

Table of n, a(n) for n=0..23.

Eric Weisstein, Horadam Sequence

Eric Weisstein, Fibonacci Number

Eric Weisstein, Pell Number

Eric Weisstein, Lucas Number

Eric Weisstein, Lucas Sequence

Index entries for linear recurrences with constant coefficients, signature (3,9).

Formula

a(n) = s*a(n-1) + r*a(n-2); for n > 3, where a(0) = 0, a(1) = 1, a(2) = 18, a(4) = 81, s = 3, r = 9.

G.f.: x*(1+15*x+18*x^2)/(1-3*x-9*x^2). [Colin Barker, Jun 20 2012]

Example

a(4) = 405 because a(3) = 81, a(2) = 18, s = 3, r = 9 and (3 * 81) + (9 * 18) = 405.

Mathematica

Join[{0, 1}, LinearRecurrence[{3, 9}, {18, 81}, 30]] (* or *) CoefficientList[ Series[x (1+15x+18x^2)/(1-3x-9x^2), {x, 0, 30}], x] (* Harvey P. Dale, Nov 24 2012 *)

Crossrefs

Cf. A024318, A000032, A000129, A001076, A085939.

Essentially the same as A122069 and A099012.

Sequence in context: A043231 A044011 A338039 * A214531 A271502 A235641

Adjacent sequences: A085501 A085502 A085503 * A085505 A085506 A085507

Keyword

nonn,easy

Author

Ross La Haye, Aug 18 2003

Extensions

First formula corrected and more terms from Harvey P. Dale, Nov 24 2012

Status

approved