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A157440 a(n) = 121*n^2 - 204*n + 86.

%I A157440 #29 Sep 15 2022 02:37:18
%S A157440 3,162,563,1206,2091,3218,4587,6198,8051,10146,12483,15062,17883,
%T A157440 20946,24251,27798,31587,35618,39891,44406,49163,54162,59403,64886,
%U A157440 70611,76578,82787,89238,95931,102866,110043,117462,125123,133026,141171
%N A157440 a(n) = 121*n^2 - 204*n + 86.
%C A157440 The identity (14641*n^2 - 24684*n + 10405)^2 - (121*n^2 - 204*n + 86)*(1331*n - 1122)^2 = 1 can be written as A157442(n)^2 - a(n)*A157441(n)^2 = 1. - _Vincenzo Librandi_, Jan 29 2012
%C A157440 The continued fraction expansion of sqrt(a(n)) is [11n-10; {1, 2, 1, 2, 11n-10, 2, 1, 2, 1, 22n-20}]. For n=1, this collapses to [1; {1, 2}]. - _Magus K. Chu_, Sep 13 2022
%H A157440 Vincenzo Librandi, <a href="/A157440/b157440.txt">Table of n, a(n) for n = 1..10000</a>
%H A157440 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>
%H A157440 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A157440 G.f.: x*(-3 - 153*x - 86*x^2)/(x-1)^3. - _Vincenzo Librandi_, Jan 29 2012
%F A157440 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jan 29 2012
%t A157440 LinearRecurrence[{3,-3,1},{3,162,563},50] (* _Vincenzo Librandi_, Jan 29 2012 *)
%o A157440 (PARI) a(n)=121*n^2-204*n+86 \\ _Charles R Greathouse IV_, Dec 28 2011
%o A157440 (Magma) I:=[3, 162, 563]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jan 29 2012
%Y A157440 Cf. A157441, A157442.
%K A157440 nonn,easy
%O A157440 1,1
%A A157440 _Vincenzo Librandi_, Mar 01 2009