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%I #83 Sep 08 2022 08:45:11
%S 1,4,11,22,37,56,79,106,137,172,211,254,301,352,407,466,529,596,667,
%T 742,821,904,991,1082,1177,1276,1379,1486,1597,1712,1831,1954,2081,
%U 2212,2347,2486,2629,2776,2927,3082,3241,3404,3571,3742,3917,4096,4279,4466
%N a(n) = 1 + n + 2*n^2.
%C Equals (1, 2, 3, ...) convolved with (1, 2, 4, 4, 4, ...). a(3) = 22 = (1, 2, 3, 4) dot (4, 4, 2, 1) = (4 + 8 + 6 + 4). - _Gary W. Adamson_, May 01 2009
%C a(n) is also the number of ways to place 2 nonattacking bishops on a 2 X (n+1) board. - _Vaclav Kotesovec_, Jan 29 2010
%C Partial sums are A174723. - _Wesley Ivan Hurt_, Apr 16 2016
%C Also the number of irredundant sets in the n-cocktail party graph. - _Eric W. Weisstein_, Aug 09 2017
%H G. C. Greubel, <a href="/A084849/b084849.txt">Table of n, a(n) for n = 0..5000</a>
%H W. Burrows and C. Tuffley, <a href="http://arxiv.org/abs/1502.06664">Maximising common fixtures in a round robin tournament with two divisions</a>, arXiv:1502.06664 [math.CO], 2015.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph.D. Thesis, Waterford Institute of Technology, 2011.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrredundantSet.html">Irredundant Set</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Alexander_polynomial">Alexander polynomial</a> and <a href="https://en.wikipedia.org/wiki/Seifert_surface">Seifert surface</a>. [See _Peter Bala_'s comment.]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A058331(n) + A000027(n).
%F G.f.: (1 + x + 2*x^2)/(1 - x)^3.
%F a(n) = A014105(n) + 1; A100035(a(n)) = 1. - _Reinhard Zumkeller_, Oct 31 2004
%F a(n) = ceiling((2*n + 1)^2/2) - n = A001844(n) - n. - _Paul Barry_, Jul 16 2006
%F From _Gary W. Adamson_, Oct 07 2007: (Start)
%F Row sums of triangle A131901.
%F (a(n): n >= 0) is the binomial transform of (1, 3, 4, 0, 0, 0, ...). (End)
%F Equals A134082 * [1,2,3,...]. -
%F a(n) = (1 + A000217(2*n-1) + A000217(2*n+1))/2. - _Enrique Pérez Herrero_, Apr 02 2010
%F a(n) = (A177342(n+1) - A177342(n))/2, with n > 0. - _Bruno Berselli_, May 19 2010
%F a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 0, with n > 2. - _Bruno Berselli_, May 24 2010
%F a(n) = 4*n + a(n-1) - 1 (with a(0) = 1). - _Vincenzo Librandi_, Aug 08 2010
%F With an offset of 1, the polynomial a(t-1) = 2*t^2 - 3*t + 2 is the Alexander polynomial (with negative powers cleared) of the 3-twist knot. The associated Seifert matrix S is [[-1,-1], [0,-2]]. a(n-1) = det(transpose(S) - n*S). Cf. A060884. - _Peter Bala_, Mar 14 2012
%F E.g.f.: (1 + 3*x + 2*x^2)*exp(x). - _Ilya Gutkovskiy_, Apr 16 2016
%p A084849:=n->1+n+2*n^2: seq(A084849(n), n=0..100); # _Wesley Ivan Hurt_, Apr 15 2016
%t s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 200, 4}]; lst (* _Zerinvary Lajos_, Jul 11 2009 *)
%t f[n_]:=(n*(2*n+1)+1);Table[f[n],{n,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2010 *)
%t Table[1 + n + 2 n^2, {n, 0, 20}] (* _Eric W. Weisstein_, Aug 09 2017 *)
%t LinearRecurrence[{3, -3, 1}, {4, 11, 22}, {0, 20}] (* _Eric W. Weisstein_, Aug 09 2017 *)
%t CoefficientList[Series[(-1 - x - 2 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 09 2017 *)
%o (PARI) a(n)=1+n+2*n^2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [1+n+2*n^2 : n in [0..100]]; // _Wesley Ivan Hurt_, Apr 15 2016
%Y Cf. A000027, A000217, A001844, A004767 (first differences), A014105, A058331, A060884, A100036, A100037, A100038, A100039, A100040, A100041, A131901, A134082, A174723, A177342.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jun 09 2003
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