A157734 - OEIS

%I #18 Sep 08 2022 08:45:42

%S 135,1064,2875,5568,9143,13600,18939,25160,32263,40248,49115,58864,

%T 69495,81008,93403,106680,120839,135880,151803,168608,186295,204864,

%U 224315,244648,265863,287960,310939,334800,359543,385168,411675,439064

%N a(n) = 441*n^2 - 394*n + 88.

%C The identity (388962*n^2 - 347508*n + 77617)^2 - (441*n^2 - 394*n + 88)*(18522*n - 8274)^2 = 1 can be written as A157736(n)^2 - a(n)*A157735(n)^2 = 1.

%C 441*a(n) + 1 is a square. - _Bruno Berselli_, Apr 23 2018

%H Vincenzo Librandi, <a href="/A157734/b157734.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*(135 + 659*x + 88*x^2)/(1 - x)^3.

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

%F a(n) = (9*n - 4)*(49*n - 22). - _Bruno Berselli_, Apr 23 2018

%t LinearRecurrence[{3, -3, 1}, {135, 1064, 2875}, 40]

%o (Magma) I:=[135, 1064, 2875]; [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) = 441*n^2 - 394*n + 88.

%Y Cf. A157735, A157736.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 05 2009