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A195562 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/4.

%I A195562 #16 Jun 04 2015 14:31:58
%S A195562 1,24,40,63,1600,2624,4161,105560,173160,274559,6965376,11425920,
%T A195562 18116737,459609240,753937576,1195430079,30327244480,49748454080,
%U A195562 78880268481,2001138526424,3282644031720,5204902289663,132044815499520
%N A195562 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/4.
%C A195562 See A195500 for a discussion and references.
%F A195562 Conjecture: a(n) = 65*a(n-3) + 65*a(n-6) - a(n-9). - _R. J. Mathar_, Sep 21 2011
%F A195562 Empirical g.f.: x*(x^6+24*x^5+40*x^4-2*x^3+40*x^2+24*x+1) / (x^9-65*x^6-65*x^3+1). - _Colin Barker_, Jun 04 2015
%t A195562 Remove["Global`*"];
%t A195562 r = 1/4; z = 26;
%t A195562 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195562       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195562          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195562      Array[FromContinuedFraction[
%t A195562         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195562 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195562   p[{r, z}]]  (* A195562, A195563 *)
%t A195562 Sqrt[a^2 + b^2] (* A195564 *)
%t A195562 (* _Peter J. C. Moses_, Sep 02 2011 *)
%Y A195562 Cf. A195500, A195563, A195564.
%K A195562 nonn
%O A195562 1,2
%A A195562 _Clark Kimberling_, Sep 21 2011