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A195614 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.

%I A195614 #41 Oct 13 2021 21:03:29
%S A195614 8,136,2448,43920,788120,14142232,253772064,4553754912,81713816360,
%T A195614 1466294939560,26311595095728,472142416783536,8472251907007928,
%U A195614 152028391909359160,2728038802461456960,48952670052396866112,878420022140682133064
%N A195614 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.
%C A195614 See A195500 for a discussion and references.
%H A195614 Colin Barker, <a href="/A195614/b195614.txt">Table of n, a(n) for n = 1..797</a>
%H A195614 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,-1).
%F A195614 From _Colin Barker_, Jun 04 2015: (Start)
%F A195614 G.f.: 8*x / ((x+1)*(x^2-18*x+1)).
%F A195614 a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3). (End)
%F A195614 a(n) = (-4*(-1)^n - (-2+sqrt(5))*(9+4*sqrt(5))^(-n) + (2+sqrt(5))*(9+4*sqrt(5))^n)/10. - _Colin Barker_, Mar 04 2016
%F A195614 a(n) = A014445(n) * A014445(n+1) / 2. - _Diego Rattaggi_, Jun 01 2020
%F A195614 a(n) is the numerator of continued fraction [4, ..., 4, 8, 4, ..., 4] with (n-1) 4's before and after the middle 8. - _Greg Dresden_ and _Hexuan Wang_, Aug 30 2021
%t A195614 r = 2; z = 32;
%t A195614 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195614       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195614          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195614      Array[FromContinuedFraction[
%t A195614         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195614 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195614   p[{r, z}]]  (* A195614, A195615 *)
%t A195614 Sqrt[a^2 + b^2] (* A007805 *)
%t A195614 (* _Peter J. C. Moses_, Sep 02 2011 *)
%o A195614 (PARI) Vec(8*x/((x+1)*(x^2-18*x+1)) + O(x^50)) \\ _Colin Barker_, Jun 04 2015
%Y A195614 Cf. A007805, A014445, A195500, A195615.
%K A195614 nonn,easy
%O A195614 1,1
%A A195614 _Clark Kimberling_, Sep 22 2011