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
KEYWORD
easy,nonn
AUTHOR
Ross La Haye, Aug 16 2003
STATUS
approved