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A195616 Denominators of Pythagorean approximations to 3.

%I A195616 #23 Feb 16 2023 14:52:29
%S A195616 12,444,16872,640680,24328980,923860548,35082371856,1332206269968,
%T A195616 50588755886940,1921040517433740,72948950906595192,
%U A195616 2770139093933183544,105192336618554379492,3994538652411133237140,151687276455004508631840
%N A195616 Denominators of Pythagorean approximations to 3.
%C A195616 See A195500 for a discussion and references.
%H A195616 Colin Barker, <a href="/A195616/b195616.txt">Table of n, a(n) for n = 1..633</a>
%H A195616 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (37,37,-1).
%F A195616 From _Colin Barker_, Jun 04 2015: (Start)
%F A195616 a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3).
%F A195616 G.f.: 12*x / ((1+x)*(1-38*x+x^2)). (End)
%F A195616 From _G. C. Greubel_, Feb 13 2023: (Start)
%F A195616 a(n) = (3/10)*(A097314(n) + (-1)^n).
%F A195616 a(n) = (1/20)*(A085447(2*n+1) - 6*(-1)^n). (End)
%t A195616 r = 3; z = 20;
%t A195616 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195616       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195616          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195616      Array[FromContinuedFraction[
%t A195616         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195616 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195616   p[{r, z}]]  (* A195616, A195617 *)
%t A195616 Sqrt[a^2 + b^2] (* A097315 *)
%t A195616 (* _Peter J. C. Moses_, Sep 02 2011 *)
%t A195616 Table[(1/20)*(LucasL[2*n+1,6] -6*(-1)^n), {n,40}] (* _G. C. Greubel_, Feb 13 2023 *)
%o A195616 (PARI) Vec(12*x/((1+x)*(1-38*x+x^2)) + O(x^20)) \\ _Colin Barker_, Jun 04 2015
%o A195616 (Magma) I:=[12, 444, 16872]; [n le 3 select I[n] else 37*Self(n-1) +37*Self(n-2) -Self(n-3): n in [1..40]]; // _G. C. Greubel_, Feb 13 2023
%o A195616 (SageMath)
%o A195616 A085447=BinaryRecurrenceSequence(6,1,2,6)
%o A195616 [(A085447(2*n+1) - 6*(-1)^n)/20 for n in range(1,41)] # _G. C. Greubel_, Feb 13 2023
%Y A195616 Cf. A085447, A097314, A097315, A195500, A195617.
%K A195616 nonn,easy,frac
%O A195616 1,1
%A A195616 _Clark Kimberling_, Sep 22 2011