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A195556 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/3.

%I A195556 #19 Jun 04 2015 14:29:29
%S A195556 1,12,24,35,468,900,1333,17760,34188,50615,674424,1298232,1922041,
%T A195556 25610340,49298640,72986939,972518508,1872050076,2771581645,
%U A195556 36930092952,71088604260,105247115567,1402371013680,2699494911792,3996618809905
%N A195556 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/3.
%C A195556 See A195500 for a discussion and references.
%F A195556 Conjecture: a(n) = 37*a(n-3) + 37*a(n-6) - a(n-9). - _R. J. Mathar_, Sep 21 2011
%F A195556 Empirical g.f.: x*(x^6+12*x^5+24*x^4-2*x^3+24*x^2+12*x+1) / (x^9-37*x^6-37*x^3+1). - _Colin Barker_, Jun 04 2015
%t A195556 r = 1/3; z = 27;
%t A195556 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195556       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195556          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195556      Array[FromContinuedFraction[
%t A195556         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195556 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195556   p[{r, z}]]  (* A195556, A195557 *)
%t A195556 Sqrt[a^2 + b^2] (* A195558 *)
%t A195556 (* _Peter J. C. Moses_, Sep 02 2011 *)
%Y A195556 Cf. A195500, A195557, A195558.
%K A195556 nonn
%O A195556 1,2
%A A195556 _Clark Kimberling_, Sep 21 2011