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%I #16 Feb 14 2023 14:58:34
%S 15,273,4895,87841,1576239,28284465,507544127,9107509825,163427632719,
%T 2932589879121,52623190191455,944284833567073,16944503814015855,
%U 304056783818718321,5456077604922913919,97905340104793732225,1756840044281364266127
%N Numerators b(n) of Pythagorean approximations b(n)/a(n) to 2.
%C See A195500 for discussion and list of related sequences; see A195614 for Mathematica program.
%H Colin Barker, <a href="/A195615/b195615.txt">Table of n, a(n) for n = 1..797</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,-1).
%F From _Colin Barker_, Jun 04 2015: (Start)
%F a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3).
%F G.f.: x*(15 + 18*x - x^2)/((1+x)*(1-18*x+x^2)). - _Colin Barker_, Jun 04 2015
%F a(n) = ((-1)^n + (2+sqrt(5))*(9+4*sqrt(5))^n + (2-sqrt(5))*(9+4*sqrt(5))^(-n))/5. - _Colin Barker_, Mar 04 2016
%F From _G. C. Greubel_, Feb 13 2023: (Start)
%F a(n) = Fibonacci(3*n+1)*Fibonacci(3*n+2).
%F a(n) = (1/5)*(4*A049629(n) + (-1)^n - 5*[n=0]). (End)
%t LinearRecurrence[{17,17,-1}, {15,273,4895}, 40] (* _G. C. Greubel_, Feb 13 2023 *)
%o (PARI) Vec(x*(15+18*x-x^2)/((1+x)*(1-18*x+x^2)) + O(x^50)) \\ _Colin Barker_, Jun 04 2015
%o (Magma) [Fibonacci(3*n+1)*Fibonacci(3*n+2): n in [1..40]]; // _G. C. Greubel_, Feb 13 2023
%o (SageMath) [fibonacci(3*n+1)*fibonacci(3*n+2) for n in range(1,41)] # _G. C. Greubel_, Feb 13 2023
%Y Cf. A000045, A007805, A049629, A195500, A195614.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Sep 22 2011
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