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144*n^2 - n.
2

%I #27 Sep 08 2022 08:45:41

%S 143,574,1293,2300,3595,5178,7049,9208,11655,14390,17413,20724,24323,

%T 28210,32385,36848,41599,46638,51965,57580,63483,69674,76153,82920,

%U 89975,97318,104949,112868,121075,129570,138353,147424,156783,166430

%N 144*n^2 - n.

%C The identity (288*n-1)^2-(144*n^2-n)*(24)^2=1 can be written as A157997(n)^2-a(n)*(24)^2=1.

%H Vincenzo Librandi, <a href="/A156635/b156635.txt">Table of n, a(n) for n = 1..10000</a>

%H E. J. Barbeau, <a href="http://www.math.toronto.edu/barbeau/home.html">Polynomial Excursions</a>, Chapter 10: <a href="http://www.math.toronto.edu/barbeau/hxpol10.pdf">Diophantine equations</a> (2010), pages 84-85 (row 14 in the first table at p. 85, case d(t) = t*(12^2*t-1)).

%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*(-143-145*x)/(x-1)^3.

%t LinearRecurrence[{3,-3,1},{143,574,1293},50]

%o (Magma) I:=[143, 574, 1293]; [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)=144*n^2-n \\ _Charles R Greathouse IV_, Dec 23 2011

%Y Cf. A157997.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Feb 15 2009