%I #26 Jul 29 2025 18:43:27

%S 218295,749736,1281177,1812618,2344059,2875500,3406941,3938382,

%T 4469823,5001264,5532705,6064146,6595587,7127028,7658469,8189910,

%U 8721351,9252792,9784233,10315674,10847115,11378556,11909997,12441438,12972879

%N a(n) = 531441*n - 313146.

%C The identity (43046721*n^2-50729652*n+14945959)^2-(6561*n^2-7732*n+2278)*(531441*n-313146)^2=1 can be written as A157378(n)^2-A157376(n)*a(n)^2=1.

%H Vincenzo Librandi, <a href="/A157377/b157377.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5773147&amp;tstart=0">X^2-AY^2=1</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

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

%F G.f.: x*(218295+313146*x)/(x-1)^2.

%F E.g.f.: 81*((6561*x - 3866)*exp(x) + 3866). - _G. C. Greubel_, Feb 04 2018

%p A157377:=n->531441*n - 313146: seq(A157377(n), n=1..40); # _Wesley Ivan Hurt_, Jan 19 2017

%t LinearRecurrence[{2,-1},{218295,749736},40]

%t Table[531441*n-313146, {n,1,30}] (* _G. C. Greubel_, Feb 04 2018 *)

%o (Magma) I:=[218295, 749736]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

%o (PARI) a(n) = 531441*n - 313146

%Y Cf. A157376, A157378.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Feb 28 2009