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%I #15 Feb 14 2023 14:58:03
%S 35,1333,50615,1922041,72986939,2771581645,105247115567,3996618809905,
%T 151766267660819,5763121552301221,218846852719785575,
%U 8310417281799550633,315577009855663138475,11983615957233399711421,455061829365013525895519
%N Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3.
%C See A195500 for discussion and list of related sequences; see A195616 for Mathematica program.
%H Colin Barker, <a href="/A195617/b195617.txt">Table of n, a(n) for n = 1..632</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (37,37,-1).
%F From _Colin Barker_, Jun 04 2015: (Start)
%F a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3).
%F G.f.: x*(35+38*x-x^2) / ((1+x)*(1-38*x+x^2)). (End)
%F a(n) = (1/20)*(3*A085447(2*n+1) + 2*(-1)^n). - _G. C. Greubel_, Feb 13 2023
%t Table[(3*LucasL[2*n+1,6] +2*(-1)^n)/20, {n, 40}] (* _G. C. Greubel_, Feb 13 2023 *)
%o (PARI) Vec(-x*(x^2-38*x-35)/((x+1)*(x^2-38*x+1)) + O(x^50)) \\ _Colin Barker_, Jun 04 2015
%o (Magma) I:=[35, 1333, 50615]; [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 (SageMath)
%o A085447=BinaryRecurrenceSequence(6,1,2,6)
%o [(3*A085447(2*n+1) + 2*(-1)^n)/20 for n in range(1,41)] # _G. C. Greubel_, Feb 13 2023
%Y Cf. A085447, A097315, A195500, A195616.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Sep 22 2011
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