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A158481 a(n) = 49*n^2 + 7.

%I A158481 #27 Jan 16 2025 06:58:21
%S A158481 56,203,448,791,1232,1771,2408,3143,3976,4907,5936,7063,8288,9611,
%T A158481 11032,12551,14168,15883,17696,19607,21616,23723,25928,28231,30632,
%U A158481 33131,35728,38423,41216,44107,47096,50183,53368,56651,60032,63511,67088,70763,74536,78407
%N A158481 a(n) = 49*n^2 + 7.
%C A158481 The identity (14*n^2 + 1)^2 - (49*n^2 + 7)*(2*n)^2 = 1 can be written as A158482(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158481 Vincenzo Librandi, <a href="/A158481/b158481.txt">Table of n, a(n) for n = 1..10000</a>
%H A158481 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 A158481 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158481 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158481 G.f: 7*x*(8+5*x+x^2)/(1-x)^3.
%F A158481 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158481 Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(7))*Pi/sqrt(7) - 1)/14.
%F A158481 Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(7))*Pi/sqrt(7))/14. (End)
%F A158481 From _Elmo R. Oliveira_, Jan 15 2025: (Start)
%F A158481 E.g.f.: 7*(exp(x)*(7*x^2 + 7*x + 1) - 1).
%F A158481 a(n) = 7*A247541(n). (End)
%t A158481 LinearRecurrence[{3,-3,1},{56,203,448},40]
%o A158481 (Magma) I:=[56, 203, 448];[n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
%o A158481 (PARI) a(n)=49*n^2+7.
%Y A158481 Cf. A005843, A158482, A247541.
%K A158481 nonn,easy
%O A158481 1,1
%A A158481 _Vincenzo Librandi_, Mar 20 2009