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A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2. 0
46, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A001622.
Cf. continued fraction expansion of phi^k: A000012 (k = 1), A054977 (k = 2), A010709 (k = 3), A176260 (k = 4, for n>0), A010850 (k = 5), A040071 (k = 6, for n>0), A010868 (k = 7), this sequence (k = 8).
Sequence in context: A023934 A022076 A055766 * A261513 A036204 A270814
KEYWORD
nonn,cofr,easy
AUTHOR
Ilya Gutkovskiy, Jan 13 2016
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)