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

Author

Ross La Haye, Aug 16 2003

Keywords

• easy
• nonn

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.

Examples

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

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).

Crossrefs

Cf. A024318, A000032, A000129.

Programs

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

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 *)

PARI

x='x+O('x^30); concat([0], Vec(x/(1-4*x-6*x^2))) \\ G. C. Greubel, Jan 16 2018

Sage

[lucas_number1(n,4,-6) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

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