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
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
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