This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342710 #36 Sep 02 2025 21:55:38
%S A342710 3,18,123,843,5778,39603,271443,1860498,12752043,87403803,599074578,
%T A342710 4106118243,28143753123,192900153618,1322157322203,9062201101803,
%U A342710 62113250390418,425730551631123,2918000611027443,20000273725560978,137083915467899403,939587134549734843
%N A342710 Solutions x to the Pell-Fermat equation x^2 - 5*y^2 = 4.
%C A342710 This Pell equation is used to find the 12-gonal square numbers (see A342709).
%C A342710 The corresponding solutions y are in A033890.
%C A342710 Essentially the same as A246453. - _R. J. Mathar_, Mar 24 2021
%H A342710 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%F A342710 a(n) = 7*a(n-1) - a(n-2).
%F A342710 a(n) = 2*T(2*n+1, 3/2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Jul 02 2022
%F A342710 From _Stefano Spezia_, Apr 14 2025: (Start)
%F A342710 G.f.: 3*(1 - x)/(1 - 7*x + x^2).
%F A342710 E.g.f.: exp(7*x/2)*(3*cosh(3*sqrt(5)*x/2) + sqrt(5)*sinh(3*sqrt(5)*x/2)). (End)
%F A342710 a(n) = 2*cos((2*n+1)*arccos(3/2)). - _Eric W. Weisstein_, Sep 02 2025
%e A342710 a(1)^2 - 5 * A033890(1)^2 = 18^2 - 5 * 8^2 = 4.
%t A342710 LinearRecurrence[{7, -1}, {3, 18}, 20] (* _Amiram Eldar_, Mar 19 2021 *)
%t A342710 Table[2 ChebyshevT[2 n + 1, 3/2], {n, 0, 20}] (* _Eric W. Weisstein_, Sep 02 2025 *)
%t A342710 Table[2 Cos[(2 n + 1) ArcCos[3/2]], {n, 0, 20}] // FunctionExpand (* _Eric W. Weisstein_, Sep 02 2025 *)
%Y A342710 Cf. A033890, A342709.
%Y A342710 a(n) = 3*A049685(n). - _Hugo Pfoertner_, Mar 19 2021
%K A342710 nonn,easy,changed
%O A342710 0,1
%A A342710 _Bernard Schott_, Mar 19 2021