A085939 Horadam sequence (0,1,6,4).

0, 1, 4, 22, 112, 580, 2992, 15448, 79744, 411664, 2125120, 10970464, 56632576, 292353088, 1509207808, 7790949760, 40219045888, 207621882112, 1071801803776, 5532938507776, 28562564853760

Offset

0,3

Comments

a(n) / a(n-1) converges to sqrt(10) + 2 as n approaches infinity; sqrt(10) + 2 can also be written as sqrt(2) * (sqrt(2) + sqrt(5)), 2 * sqrt(2) * Phi - sqrt(2) + 2 and lim_{n->infinity} sqrt(2) * (sqrt(2) + (L(n) / F(n))), where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein, Lucas Number

Eric Weisstein, Lucas Sequence

Eric Weisstein, Horadam Sequence

Eric Weisstein, Fibonacci Number

Eric Weisstein, Pell Number

Index entries for linear recurrences with constant coefficients, signature (4, 6).

Formula

a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 6.

a(n) = ((2+sqrt(10))^n - (2-sqrt(10))^n)/(2*sqrt(10)). - Rolf Pleisch, Jul 06 2009

G.f.: x/(1-4*x-6*x^2). - Colin Barker, Jan 10 2012

Example

a(4) = 112 because a(3) = 22, a(2) = 4, s = 4, r = 6 and (4 * 22) + (6 * 4) = 112.

Mathematica

Join[{a=0, b=1}, Table[c=4*b+6*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{4, 6}, {0, 1}, 30] (* Harvey P. Dale, Jul 20 2016 *)

Prog

(Sage) [lucas_number1(n, 4, -6) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009
(PARI) x='x+O('x^30); concat([0], Vec(x/(1-4*x-6*x^2))) \\ G. C. Greubel, Jan 16 2018
(Magma) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Crossrefs

Cf. A024318, A000032, A000129.

Sequence in context: A144047 A077543 A084157 * A192470 A274799 A290591

Adjacent sequences: A085936 A085937 A085938 * A085940 A085941 A085942

Keyword

easy,nonn

Author

Ross La Haye, Aug 16 2003

Status

approved