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A158773 a(n) = 1600*n^2 - 40.

%I A158773 #29 Jan 27 2025 21:15:41
%S A158773 1560,6360,14360,25560,39960,57560,78360,102360,129560,159960,193560,
%T A158773 230360,270360,313560,359960,409560,462360,518360,577560,639960,
%U A158773 705560,774360,846360,921560,999960,1081560,1166360,1254360,1345560,1439960,1537560,1638360,1742360
%N A158773 a(n) = 1600*n^2 - 40.
%C A158773 The identity (80*n^2 - 1)^2 - (1600*n^2 - 40)*(2*n)^2 = 1 can be written as A158774(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158773 Vincenzo Librandi, <a href="/A158773/b158773.txt">Table of n, a(n) for n = 1..10000</a>
%H A158773 Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%H A158773 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158773 G.f.: 40*x*(-39 - 42*x + x^2)/(x-1)^3.
%F A158773 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158773 From _Amiram Eldar_, Mar 24 2023: (Start)
%F A158773 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)))/80.
%F A158773 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) - 1)/80. (End)
%F A158773 From _Elmo R. Oliveira_, Jan 26 2025: (Start)
%F A158773 E.g.f.: 40*(exp(x)*(40*x^2 + 40*x - 1) + 1).
%F A158773 a(n) = 40*A158598(n). (End)
%t A158773 LinearRecurrence[{3, -3, 1}, {1560, 6360, 14360}, 50] (* _Vincenzo Librandi_, Feb 21 2012 *)
%t A158773 1600*Range[30]^2-40 (* _Harvey P. Dale_, Apr 30 2017 *)
%o A158773 (Magma) I:=[1560, 6360, 14360]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 21 2012
%o A158773 (PARI) for(n=1, 40, print1(1600*n^2 - 40", ")); \\ _Vincenzo Librandi_, Feb 21 2012
%Y A158773 Cf. A005843, A158598, A158774.
%K A158773 nonn,easy
%O A158773 1,1
%A A158773 _Vincenzo Librandi_, Mar 26 2009
%E A158773 Comment rewritten and formula replaced by _R. J. Mathar_, Oct 22 2009