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

0,2

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.

Wikipedia, Alexander polynomial and Seifert surface. [See Peter Bala's comment.]

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

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 *)

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

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

Sequence in context: A038414 A008154 A008162 * A008265 A160424 A008229

Adjacent sequences: A084846 A084847 A084848 * A084850 A084851 A084852

Keyword

easy,nonn

Author

Paul Barry, Jun 09 2003

Status

approved