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A158484 a(n) = 49*n^2 - 7.

%I A158484 #27 Jan 28 2025 08:39:22
%S A158484 42,189,434,777,1218,1757,2394,3129,3962,4893,5922,7049,8274,9597,
%T A158484 11018,12537,14154,15869,17682,19593,21602,23709,25914,28217,30618,
%U A158484 33117,35714,38409,41202,44093,47082,50169,53354,56637,60018,63497,67074,70749,74522,78393
%N A158484 a(n) = 49*n^2 - 7.
%C A158484 The identity (14*n^2 - 1)^2 - (49*n^2 - 7)*(2*n)^2 = 1 can be written as A158485(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158484 Vincenzo Librandi, <a href="/A158484/b158484.txt">Table of n, a(n) for n = 1..10000</a>
%H A158484 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 A158484 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158484 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158484 G.f: 7*x*(-6-9*x+x^2)/(x-1)^3.
%F A158484 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158484 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(7))*Pi/sqrt(7))/14.
%F A158484 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(7))*Pi/sqrt(7) - 1)/14. (End)
%F A158484 E.g.f.: 7*(exp(x)*(7*x^2 + 7*x - 1) + 1). - _Elmo R. Oliveira_, Jan 27 2025
%t A158484 LinearRecurrence[{3,-3,1},{42,189,434},50]
%o A158484 (Magma) I:=[42, 189, 434]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
%o A158484 (PARI) a(n) = 49*n^2-7;
%Y A158484 Cf. A005843, A158485.
%K A158484 nonn,easy
%O A158484 1,1
%A A158484 _Vincenzo Librandi_, Mar 20 2009