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A158482 a(n) = 14*n^2 + 1.

%I A158482 #36 Jan 20 2024 07:09:44
%S A158482 15,57,127,225,351,505,687,897,1135,1401,1695,2017,2367,2745,3151,
%T A158482 3585,4047,4537,5055,5601,6175,6777,7407,8065,8751,9465,10207,10977,
%U A158482 11775,12601,13455,14337,15247,16185,17151,18145,19167,20217,21295,22401
%N A158482 a(n) = 14*n^2 + 1.
%C A158482 The identity (14*n^2 + 1)^2 - (49*n^2 + 7)*(2*n)^2 = 1 can be written as a(n)^2 - A158481(n)*A005843(n)^2 = 1.
%C A158482 Sequence found by reading the line from 15, in the direction 15, 57, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - _Omar E. Pol_, Sep 13 2011
%H A158482 Vincenzo Librandi, <a href="/A158482/b158482.txt">Table of n, a(n) for n = 1..10000</a>
%H A158482 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158482 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158482 G.f: x*(15+12*x+x^2)/(1-x)^3.
%F A158482 From _Amiram Eldar_, Feb 05 2021: (Start)
%F A158482 Sum_{n>=0} 1/a(n) = (1 - (Pi/sqrt(14))*coth(Pi/sqrt(14)))/2.
%F A158482 Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(14))*csch(Pi/sqrt(14)))/2.
%F A158482 Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(14))*sinh(Pi/sqrt(7)).
%F A158482 Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(14))*csch(Pi/sqrt(14)). (End)
%t A158482 LinearRecurrence[{3,-3,1},{15,57,127},50]
%o A158482 (Magma) I:=[15, 57, 127]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
%o A158482 (PARI) a(n) = 14*n^2+1;
%Y A158482 Cf. A005843, A118277, A158481, A195019, A195020.
%K A158482 nonn,easy
%O A158482 1,1
%A A158482 _Vincenzo Librandi_, Mar 20 2009