cp's OEIS Frontend

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

%I A267319 #21 Feb 16 2025 08:33:29
%S A267319 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 A267319 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 A267319 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 A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.
%C A267319 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 A267319 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A267319 Wikipedia, <a href="http://www.wikipedia.org/wiki/Golden_ratio">Golden ratio</a>
%H A267319 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A267319 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F A267319 G.f.: (46 + x - x^2)/(1 - x^2).
%F A267319 a(n) = 23 + 22*(-1)^n for n>0. - _Bruno Berselli_, Jan 18 2016
%e A267319 phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).
%t A267319 ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]
%o A267319 (Magma) [46] cat &cat [[1, 45]^^50]; // _Vincenzo Librandi_, Jan 13 2016
%Y A267319 Cf. A001622.
%Y A267319 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 A267319 nonn,cofr,easy
%O A267319 0,1
%A A267319 _Ilya Gutkovskiy_, Jan 13 2016