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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

Table of n, a(n) for n=0..81.

Wikipedia, Golden ratio

Eric Weisstein's World of Mathematics, Golden Ratio

Index entries for continued fractions for constants

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 October 22 18:55 EDT 2018. Contains 316500 sequences. (Running on oeis4.)