%I #34 May 07 2024 05:50:33
%S 1,21,78,173,306,477,686,933,1218,1541,1902,2301,2738,3213,3726,4277,
%T 4866,5493,6158,6861,7602,8381,9198,10053,10946,11877,12846,13853,
%U 14898,15981,17102,18261,19458,20693,21966,23277,24626,26013,27438,28901,30402,31941
%N a(0) = 1, a(n) = 19*n^2 + 2 for n>0.
%C Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=1, s=3. After 1, all terms are in A000408. - _Bruno Berselli_, Feb 06 2012
%H Bruno Berselli, <a href="/A010009/b010009.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1+x)*(1+17*x+x^2)/(1-x)^3. - _Bruno Berselli_, Feb 06 2012
%F E.g.f.: (x*(x+1)*19+2)*e^x-1. - _Gopinath A. R._, Feb 14 2012
%F Sum_{n>=0} 1/a(n) = 3/4 + sqrt(38)/76*Pi*coth(Pi*sqrt(38)/19) = 1.08111673149128.. - _R. J. Mathar_, May 07 2024
%F a(n) = A069132(n)+A069132(n+1). - _R. J. Mathar_, May 07 2024
%t Join[{1}, 19 Range[41]^2 + 2] (* _Harvey P. Dale_, Feb 07 2011 *)
%t Join[{1}, LinearRecurrence[{3, -3, 1}, {21, 78, 173}, 50]] (* _Vincenzo Librandi_, Aug 03 2015 *)
%o (Magma) [1] cat [19*n^2+2: n in [1..50]]; // _Vincenzo Librandi_, Aug 03 2015
%Y Cf. A206399.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_.