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 A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2. 0

%I

%S 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,

%T 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,

%U 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

%N Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

%C 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).

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Golden_ratio">Golden ratio</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).

%F G.f.: (46 + x - x^2)/(1 - x^2).

%F a(n) = 23 + 22*(-1)^n for n>0. [_Bruno Berselli_, Jan 18 2016]

%e phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).

%t ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]

%o (MAGMA) [46] cat &cat [[1, 45]^^50]; // _Vincenzo Librandi_, Jan 13 2016

%Y Cf. A001622.

%Y 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).

%K nonn,cofr,easy

%O 0,1

%A _Ilya Gutkovskiy_, Jan 13 2016

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Last modified October 16 14:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)