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A195619 Denominators of Pythagorean approximations to 4.

%I A195619 #32 May 01 2023 17:27:53
%S A195619 16,1040,68640,4529184,298857520,19720067120,1301225572416,
%T A195619 85861167712320,5665535843440720,373839504499375184,
%U A195619 24667741761115321440,1627697116729111839840,107403341962360266108016,7086992872399048451289200
%N A195619 Denominators of Pythagorean approximations to 4.
%C A195619 See A195500 for a discussion and references.
%H A195619 Colin Barker, <a href="/A195619/b195619.txt">Table of n, a(n) for n = 1..549</a>
%H A195619 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (65,65,-1).
%F A195619 From _Colin Barker_, Jun 03 2015: (Start)
%F A195619 a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
%F A195619 G.f.: 16*x/((1+x)*(1-66*x+x^2)). (End)
%F A195619 a(n) = ((4+sqrt(17))^(2*n+1) + (4-sqrt(17))^(2*n+1) - 8*(-1)^n)/34. - _Colin Barker_, Mar 03 2016
%F A195619 a(n) = 4*(A078989(n) - (-1)^n)/17. - _G. C. Greubel_, Feb 13 2023
%t A195619 r = 4; z = 20;
%t A195619 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195619       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195619          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195619      Array[FromContinuedFraction[
%t A195619         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195619 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195619   p[{r, z}]]  (* A195619, A195620 *)
%t A195619 Sqrt[a^2 + b^2] (* A078988 *)
%t A195619 (* _Peter J. C. Moses_, Sep 02 2011 *)
%t A195619 Table[(LucasL[2*n+1,8] - 8*(-1)^n)/34, {n,40}] (* _G. C. Greubel_, Feb 13 2023 *)
%t A195619 LinearRecurrence[{65,65,-1},{16,1040,68640},20] (* _Harvey P. Dale_, May 01 2023 *)
%o A195619 (PARI) Vec(16*x/((x+1)*(x^2-66*x+1)) + O(x^20)) \\ _Colin Barker_, Jun 03 2015
%o A195619 (Magma) I:=[16, 1040, 68640]; [n le 3 select I[n] else 65*Self(n-1) +65*Self(n-2) -Self(n-3): n in [1..40]]; // _G. C. Greubel_, Feb 13 2023
%o A195619 (SageMath)
%o A195619 A078989=BinaryRecurrenceSequence(66,-1,1,67)
%o A195619 [4*(A078989(n) - (-1)^n)/17 for n in range(1,41)] # _G. C. Greubel_, Feb 13 2023
%Y A195619 Cf. A078988, A078989, A195500, A195620.
%K A195619 nonn,easy,frac
%O A195619 1,1
%A A195619 _Clark Kimberling_, Sep 22 2011