A084849 a(n) = 1 + n + 2*n^2.

1, 4, 11, 22, 37, 56, 79, 106, 137, 172, 211, 254, 301, 352, 407, 466, 529, 596, 667, 742, 821, 904, 991, 1082, 1177, 1276, 1379, 1486, 1597, 1712, 1831, 1954, 2081, 2212, 2347, 2486, 2629, 2776, 2927, 3082, 3241, 3404, 3571, 3742, 3917, 4096, 4279, 4466

Offset: 0

Author

Paul Barry, Jun 09 2003

Keywords

• easy
• nonn

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.
• Wikipedia, Alexander polynomial and Seifert surface. [See _Peter Bala_'s comment.]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000027, A000217, A001844, A004767 (first differences), A014105, A058331, A060884, A100036, A100037, A100038, A100039, A100040, A100041, A131901, A134082, A174723, A177342.

Programs

Magma

[1+n+2*n^2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016

Maple

A084849:=n->1+n+2*n^2: seq(A084849(n), n=0..100); # Wesley Ivan Hurt, Apr 15 2016

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 *)

PARI

a(n)=1+n+2*n^2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = A058331(n) + A000027(n).

G.f.: (1 + x + 2*x^2)/(1 - x)^3.

a(n) = A014105(n) + 1; A100035(a(n)) = 1. - Reinhard Zumkeller, Oct 31 2004

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) = (1 + A000217(2*n-1) + A000217(2*n+1))/2. - Enrique Pérez Herrero, Apr 02 2010

a(n) = (A177342(n+1) - A177342(n))/2, with n > 0. - Bruno Berselli, May 19 2010

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