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

Author

Ross La Haye, Aug 18 2003

Keywords

• nonn
• easy

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)).

Examples

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

Links

• 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).

Crossrefs

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

Essentially the same as A122069 and A099012.

Programs

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 *)

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]

Extensions

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