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a(n) = 388962*n^2 - 430416*n + 119071.
3

%I #23 Sep 08 2022 08:45:42

%S 77617,814087,2328481,4620799,7691041,11539207,16165297,21569311,

%T 27751249,34711111,42448897,50964607,60258241,70329799,81179281,

%U 92806687,105212017,118395271,132356449,147095551,162612577,178907527

%N a(n) = 388962*n^2 - 430416*n + 119071.

%C The identity (388962*n^2 - 430416*n + 119071)^2 - (441*n^2 - 488*n + 135)*(18522*n - 10248)^2 = 1 can be written as a(n)^2 - A157730(n)*A157731(n)^2 = 1.

%C This is the case s=21 and r=244 in the identity (2*(s^2*n-r)^2-1)^2 - (((s^2*n-r)^2-1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2-1)/s^2 is an integer if r^2 == 1 (mod s^2). - _Bruno Berselli_, Apr 23 2018

%H Vincenzo Librandi, <a href="/A157732/b157732.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: x*(77617 + 581236*x + 119071*x^2)/(1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%t LinearRecurrence[{3,-3, 1}, {77617, 814087, 2328481}, 40]

%o (Magma) I:=[77617, 814087, 2328481]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

%o (PARI) a(n) = 388962*n^2 - 430416*n + 119071.

%Y Cf. A157730, A157731.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 05 2009