A342710 - OEIS

%I #25 Jul 11 2022 08:35:55

%S 3,18,123,843,5778,39603,271443,1860498,12752043,87403803,599074578,

%T 4106118243,28143753123,192900153618,1322157322203,9062201101803,

%U 62113250390418,425730551631123,2918000611027443,20000273725560978,137083915467899403,939587134549734843

%N Solutions x to the Pell-Fermat equation x^2 - 5*y^2 = 4.

%C This Pell equation is used to find the 12-gonal square numbers (see A342709).

%C The corresponding solutions y are in A033890.

%C Essentially the same as A246453. - _R. J. Mathar_, Mar 24 2021

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).

%F a(n) = 7*a(n-1) - a(n-2).

%F 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

%e a(1)^2 - 5 * A033890(1)^2 = 18^2 - 5 * 8^2 = 4.

%t LinearRecurrence[{7, -1}, {3, 18}, 20] (* _Amiram Eldar_, Mar 19 2021 *)

%Y Cf. A033890, A342709.

%Y a(n) = 3*A049685(n). - _Hugo Pfoertner_, Mar 19 2021

%K nonn,easy

%O 0,1

%A _Bernard Schott_, Mar 19 2021