A157736 - OEIS

%I #22 Sep 08 2022 08:45:42

%S 119071,938449,2535751,4910977,8064127,11995201,16704199,22191121,

%T 28455967,35498737,43319431,51918049,61294591,71449057,82381447,

%U 94091761,106579999,119846161,133890247,148712257,164312191,180690049

%N a(n) = 388962*n^2 - 347508*n + 77617.

%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 a(n)^2 - A157734(n)*A157735(n)^2 = 1.

%C This is the case s=21 and r=197 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="/A157736/b157736.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*(119071 + 581236*x + 77617*x^2)/(1 - x)^3.

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

%t LinearRecurrence[{3, -3, 1}, {119071, 938449, 2535751}, 40]

%o (Magma) I:=[119071, 938449, 2535751]; [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 - 347508*n + 77617.

%Y Cf. A157734, A157735.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 05 2009