OFFSET
0,2
COMMENTS
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
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
Partial sums are A174723. - Wesley Ivan Hurt, Apr 16 2016
Also the number of irredundant sets in the n-cocktail party graph. - Eric W. Weisstein, Aug 09 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
W. Burrows and C. Tuffley, Maximising common fixtures in a round robin tournament with two divisions, arXiv:1502.06664 [math.CO], 2015.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph.D. Thesis, Waterford Institute of Technology, 2011.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Irredundant Set.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (1 + x + 2*x^2)/(1 - x)^3.
a(n) = ceiling((2*n + 1)^2/2) - n = A001844(n) - n. - Paul Barry, Jul 16 2006
From Gary W. Adamson, Oct 07 2007: (Start)
Row sums of triangle A131901.
(a(n): n >= 0) is the binomial transform of (1, 3, 4, 0, 0, 0, ...). (End)
Equals A134082 * [1,2,3,...]. -
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 0, with n > 2. - Bruno Berselli, May 24 2010
a(n) = 4*n + a(n-1) - 1 (with a(0) = 1). - Vincenzo Librandi, Aug 08 2010
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
E.g.f.: (1 + 3*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Apr 16 2016
MAPLE
MATHEMATICA
s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 200, 4}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
f[n_]:=(n*(2*n+1)+1); Table[f[n], {n, 5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Table[1 + n + 2 n^2, {n, 0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{3, -3, 1}, {4, 11, 22}, {0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
CoefficientList[Series[(-1 - x - 2 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Aug 09 2017 *)
PROG
(PARI) a(n)=1+n+2*n^2 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [1+n+2*n^2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Paul Barry, Jun 09 2003
STATUS
approved