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%I #71 Feb 21 2022 10:52:55
%S 1,4,16,37,67,106,154,211,277,352,436,529,631,742,862,991,1129,1276,
%T 1432,1597,1771,1954,2146,2347,2557,2776,3004,3241,3487,3742,4006,
%U 4279,4561,4852,5152,5461,5779,6106,6442,6787,7141,7504,7876,8257,8647,9046
%N a(n) = (9*n^2 - 3*n + 2)/2.
%C The old definition of this sequence was "Generalized polygonal numbers".
%C Row T(3,n) of A080853.
%C Equals binomial transform of [1, 3, 9, 0, 0, 0, ...] - _Gary W. Adamson_, Apr 30 2008
%C a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is congruent to 2 modulo 3. The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4, are 2+2, 5+5+2+2+2, 8+8+5+5+5+2+2+2, 11+11+8+8+8+5+5+5+2+2+2. - _Augustine O. Munagi_, Dec 18 2008
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1) = -coeff(charpoly(A,x), x^(n-2)). - _Milan Janjic_, Jan 27 2010
%H Vincenzo Librandi, <a href="/A080855/b080855.txt">Table of n, a(n) for n = 0..200</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H A. O. Munagi, <a href="http://dx.doi.org/10.1016/j.disc.2007.05.022">Pairing conjugate partitions by residue classes</a>, Discrete Math., 308 (2008), 2492-2501.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H Leo Tavares, <a href="/A080855/a080855.jpg">Illustration: Hexagonal Tri-Rays</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 + 7*x^2)/(1 - x)^3.
%F a(n) = 9*n + a(n-1) - 6 with n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = n*A005448(n+1) - (n-1)*A005448(n), with A005448(0)=1. - _Bruno Berselli_, Jan 15 2013
%F a(0)=1, a(1)=4, a(2)=16; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jul 24 2013
%F a(n) = A152947(3*n+1). - _Franck Maminirina Ramaharo_, Jan 10 2018
%F E.g.f.: (2 + 6*x + 9*x^2)*exp(x)/2. - _G. C. Greubel_, Nov 02 2018
%F From _Leo Tavares_, Feb 20 2022: (Start)
%F a(n) = A003215(n-1) + 3*A000217(n). See Hexagonal Tri-Rays illustration in links.
%F a(n) = A227776(n) - 3*A000217(n). (End)
%p seq((9*n^2-3*n+2)/2,n=0..50); # _Muniru A Asiru_, Nov 02 2018
%t s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 500, 9}]; lst (* _Zerinvary Lajos_, Jul 11 2009 *)
%t Table[(9n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1}, {1,4,16}, 50] (* _Harvey P. Dale_, Jul 24 2013 *)
%o (PARI) a(n)=binomial(3*n,2)+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [(9*n^2 - 3*n +2)/2: n in [0..50]]; // _G. C. Greubel_, Nov 02 2018
%o (GAP) List([0..50],n->(9*n^2-3*n+2)/2); # _Muniru A Asiru_, Nov 02 2018
%Y Cf. A027468, A038764.
%Y Cf. A283394 (see Crossrefs section).
%Y Cf. A003215, A000217, A227776.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 23 2003
%E Definition replaced with the closed form by _Bruno Berselli_, Jan 15 2013