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a(n) = 4*n^2 + 79*n + 390.
3

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

%S 473,564,663,770,885,1008,1139,1278,1425,1580,1743,1914,2093,2280,

%T 2475,2678,2889,3108,3335,3570,3813,4064,4323,4590,4865,5148,5439,

%U 5738,6045,6360,6683,7014,7353,7700,8055,8418,8789,9168,9555,9950,10353,10764

%N a(n) = 4*n^2 + 79*n + 390.

%C The identity (128*n^2 + 2528*n + 12481)^2 - (4*n^2 + 79*n + 390)*(64*n + 632)^2 = 1 can be written as A157436(n)^2 - a(n)*A157435(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A157434/b157434.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_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F G.f.: x*(-390*x^2 + 855*x - 473)/(x-1)^3. [corrected by _Georg Fischer_, May 11 2019]

%t LinearRecurrence[{3,-3,1},{473,564,663},50]

%o (Magma) I:=[473, 564, 663]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];

%o (PARI) a(n) = 4*n^2 + 79*n + 390

%Y Cf. A157435, A157436.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 01 2009