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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 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 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 Adjacent sequences:  A267316 A267317 A267318 * A267320 A267321 A267322 KEYWORD nonn,cofr,easy AUTHOR Ilya Gutkovskiy, Jan 13 2016 STATUS approved

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Last modified January 15 21:50 EST 2021. Contains 340195 sequences. (Running on oeis4.)