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%I #19 Jul 03 2023 14:33:58
%S 137,253772063,472142416783537,878420022140682133063,
%T 1634298694352222684783778137,3040609452244043180572708973082863,
%U 5657047804679503550674811676317937783937,10524926126507566387571141730985597902165021463
%N a(n) = sqrt((A200216(n)+1)/5).
%C All numbers (A200216(n)+1)/5 are perfect squares
%H Vincenzo Librandi, <a href="/A200335/b200335.txt">Table of n, a(n) for n = 1..50</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1860497, 1860497, -1).
%F G.f.: (137 - 1116026*x + 137*x^2)/(1 - 1860497*x - 1860497*x^2 + x^3).
%F a(n) = 1860497*a(n-1) + 1860497*a(n-2) - a(n-3). [corrected by _Vincenzo Librandi_, Nov 18 2011]
%t aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, Sqrt[(xx + 1)/5]], {n, 1, 20}]; aa
%o (Magma) I:=[137, 253772063, 472142416783537]; [n le 3 select I[n] else 1860497*Self(n-1)+1860497*Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Nov 18 2011
%o (PARI) x='x+O('x^30); Vec((137 -1116026*x +137*x^2)/(1 - 1860497*x - 1860497*x^2 + x^3)) \\ _G. C. Greubel_, Jul 10 2018
%Y Cf. A200216, A200217, A200218.
%K nonn
%O 1,1
%A _Artur Jasinski_, Nov 16 2011
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