%I #30 Aug 06 2024 22:33:32
%S 19,73,163,289,451,649,883,1153,1459,1801,2179,2593,3043,3529,4051,
%T 4609,5203,5833,6499,7201,7939,8713,9523,10369,11251,12169,13123,
%U 14113,15139,16201,17299,18433,19603,20809,22051,23329,24643,25993,27379,28801,30259,31753
%N a(n) = 18*n^2 + 1.
%C The identity (18n^2 + 1)^2 - (81n^2 + 9)*(2n)^2 = 1 can be written as a(n)^2 - A157888(n)*A005843(n+1)^2 = 1. - _Vincenzo Librandi_, Feb 05 2012
%C Sequence found by reading the line from 19, in the direction 19, 73, ... in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - _Omar E. Pol_, Nov 05 2012
%H Vincenzo Librandi, <a href="/A157889/b157889.txt">Table of n, a(n) for n = 1..10000</a>
%H Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Vincenzo Librandi_, Feb 05 2012: (Start)
%F G.f: x*(19 + 16*x + x^2)/(1-x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F From _Amiram Eldar_, Mar 07 2023: (Start)
%F Sum_{n>=1} 1/a(n) = (coth(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) - 1)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/2. (End)
%t LinearRecurrence[{3, -3, 1}, {19, 73, 163}, 40] (* _Vincenzo Librandi_, Feb 05 2012 *)
%o (Magma) I:=[19, 73, 163]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Feb 05 2012
%o (PARI) for(n=1, 40, print1(18n^2 + 1", ")); \\ _Vincenzo Librandi_, Feb 05 2012
%Y Cf. A005843, A157888, A195160.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 08 2009