OFFSET
0,1
COMMENTS
More generally, the ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2, k = 1, 2, 3,... is floor(phi^(2*k + 1))/(1 - x), and for the continued fraction expansion of phi^(2*k) is (floor(phi^(2*k)) + x - x^2)/(1 - x^2).
LINKS
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden ratio
Index entries for linear recurrences with constant coefficients, signature (0,1).
FORMULA
G.f.: (46 + x - x^2)/(1 - x^2).
a(n) = 23 + 22*(-1)^n for n>0. - Bruno Berselli, Jan 18 2016
EXAMPLE
phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).
MATHEMATICA
ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]
PROG
(Magma) [46] cat &cat [[1, 45]^^50]; // Vincenzo Librandi, Jan 13 2016
CROSSREFS
KEYWORD
nonn,cofr,easy,changed
AUTHOR
Ilya Gutkovskiy, Jan 13 2016
STATUS
approved