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A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).

%I A082766 #19 Dec 15 2022 15:23:04
%S A082766 1,1,2,3,4,7,10,17,24,41,58,99,140,239,338,577,816,1393,1970,3363,
%T A082766 4756,8119,11482,19601,27720,47321,66922,114243,161564,275807,390050,
%U A082766 665857,941664,1607521,2273378,3880899,5488420,9369319,13250218,22619537
%N A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).
%C A082766 a(2n+2)/a(2n+1) converges to sqrt(2).
%C A082766 a(2n+1)/a(2n) converges to 1+sqrt(1/2).
%C A082766 a(n+2)/a(n) converges to 1+sqrt(2).
%C A082766 a(2n) is A001333, a(2n+1) is A052542.
%H A082766 Reinhard Zumkeller, <a href="/A082766/b082766.txt">Table of n, a(n) for n = 1..1000</a>
%H A082766 Haocong Song and Wen Wu, <a href="https://arxiv.org/abs/2007.09940">Hankel determinants of a Sturmian sequence</a>, arXiv:2007.09940 [math.CO], 2020. See p. 4.
%H A082766 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1).
%F A082766 a(2n) = a(2n-1) + a(2n-2); a(2n+1) = a(2n) + a(2n-2)
%F A082766 O.g.f.: x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4). - _R. J. Mathar_, Aug 08 2008
%t A082766 Rest[CoefficientList[Series[x (1 - x^2 + x) (x^2 + 1)/(1 - 2 x^2 - x^4), {x, 0, 50}], x]] (* _G. C. Greubel_, Nov 28 2017 *)
%t A082766 LinearRecurrence[{0,2,0,1},{1,1,2,3,4},50] (* _Harvey P. Dale_, Dec 15 2022 *)
%o A082766 (Haskell)
%o A082766 import Data.List (transpose)
%o A082766 a082766 n = a082766_list !! (n-1)
%o A082766 a082766_list = concat $ transpose [a052542_list, tail a001333_list]
%o A082766 -- _Reinhard Zumkeller_, Feb 24 2015
%o A082766 (PARI) x='x+O('x^30); Vec(x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4)) \\ _G. C. Greubel_, Nov 28 2017
%Y A082766 Cf. A001333, A052542. See A119016 for another version.
%K A082766 nonn
%O A082766 1,3
%A A082766 _Gary W. Adamson_, May 24 2003
%E A082766 Edited by _Don Reble_, Nov 04 2005