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A082109 Third row of number array A082105.

%I A082109 #71 Jul 21 2025 00:27:32
%S A082109 1,13,33,61,97,141,193,253,321,397,481,573,673,781,897,1021,1153,1293,
%T A082109 1441,1597,1761,1933,2113,2301,2497,2701,2913,3133,3361,3597,3841,
%U A082109 4093,4353,4621,4897,5181,5473,5773,6081,6397,6721,7053,7393,7741,8097,8461
%N A082109 Third row of number array A082105.
%H A082109 G. C. Greubel, <a href="/A082109/b082109.txt">Table of n, a(n) for n = 0..1000</a>
%H A082109 Russ Cox et al., <a href="https://groups.google.com/g/seqfan/c/zfh_LfkwCg8">incorrect A082109 comment about A000217</a>, SeqFan, 2024.
%H A082109 Takao Komatsu, Ritika Goel, and Neha Gupta, <a href="https://arxiv.org/abs/2409.14788">The Frobenius number for the triple of the 2-step star numbers</a>, arXiv:2409.14788 [math.CO], 2024. See p. 2.
%H A082109 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A082109 a(n) = 4*n^2 + 8*n + 1.
%F A082109 a(n) = a(n-1) + 8*n + 4, with a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A082109 G.f.: (1 + 10*x - 3*x^2)/(1-x)^3. - _Bruno Berselli_, Apr 18 2011
%F A082109 E.g.f.: (1 + 12*x + 4*x^2)*exp(x). - _G. C. Greubel_, Dec 22 2022
%F A082109 From _Amiram Eldar_, Jan 18 2023: (Start)
%F A082109 Sum_{n>=0} 1/a(n) = 1/6 - cot(sqrt(3)*Pi/2)*sqrt(3)*Pi/12.
%F A082109 Sum_{n>=0} (-1)^n/a(n) = cosec(sqrt(3)*Pi/2)*sqrt(3)*Pi/12 - 1/6. (End)
%t A082109 LinearRecurrence[{3,-3,1}, {1,13,33}, 51] (* _Vladimir Joseph Stephan Orlovsky_, Oct 25 2008 *)
%o A082109 (PARI) a(n)=4*n^2+8*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A082109 (Magma) [4*n^2+8*n+1: n in [0..60]]; // _G. C. Greubel_, Dec 22 2022
%o A082109 (SageMath) [4*n^2+8*n+1 for n in range(61)] # _G. C. Greubel_, Dec 22 2022
%Y A082109 Cf. A000217, A082105, A082108, A000217.
%Y A082109 Column 2 of array A188646.
%K A082109 easy,nonn
%O A082109 0,2
%A A082109 _Paul Barry_, Apr 03 2003