OFFSET

1,3

COMMENTS

The sequence is the sum of upward sloping terms in an infinite lower triangle with 1's in the leftmost column and the odd integers in all other columns. - Gary W. Adamson, Jan 29 2014

For n > 1, if Kruskal's algorithm is run on a weighted connected graph of n nodes, then a(n) is the maximum number of iterations required to reach a spanning tree. - Eric M. Schmidt, Jun 04 2016

It can be observed that A152947/A000079, whose reduced numerators are A213671, is identical to its inverse binomial transform (except for signs); this shows that it is an "autosequence" (more precisely, an autosequence of the second kind). - Jean-François Alcover (this remark is due to Paul Curtz), Jun 20 2016

Harnack's theorem (1876) states that the number of components of a plane algebraic curve of order n is at most a(n) and that this number can be achieved. For example, the zero set of a quadratic has at most 1 component (e.g. a circle); a cubic elliptic curve has at most 2 components. The number of topological arrangements (Hilbert's 16th problem) is only known for a few values of n. For quartics, a(4)=4 and there are 6 topological arrangements: 0 to 4 unnested ovals or 2 nested ovals. - Robert McLachlan, Aug 19 2024

LINKS

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Christian Bean, Bjarki Gudmundsson, and Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.

Murat Ersen Berberler, Onur Ugurlu, and Gozde Kizilates, On a Subroutine for Covering Zeros in Hungarian Algorithm, 2012, see section 5.1 on page 92.

Hayat Cheballah, Samuele Giraudo, and Rémi Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.

Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 9.

D. A. Gudkov, The topology of real projective algebraic varieties, Russ. Math. Surv. 29 (1974), 1-79.

Lara Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

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

FORMULA

a(n) = a(n-1) + n - 2 for n>1, a(1) = 1. - Vincenzo Librandi, Nov 26 2010

G.f.: -x*(1-2*x+2*x^2)/(x-1)^3. - R. J. Mathar, Nov 28 2010

From Ilya Gutkovskiy, Jun 04 2016: (Start)

E.g.f.: (4 - 2*x + x^2)*exp(x)/2 - 2.

Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) + 1 = A226985 + 1. (End)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - Wesley Ivan Hurt, Jun 20 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - A228918. - Amiram Eldar, Jan 08 2023

MAPLE

MATHEMATICA

Table[1 + (n^2 - 3n + 2)/2, {n, 50}] (* Alonso del Arte, Jan 30 2014 *)

PROG

(Sage) [1+binomial(n, 2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009

(Magma) [1+(n-2)*(n-1)/2: n in [1..60]]; // Klaus Brockhaus, Nov 28 2010

(PARI) a(n)=1+(n-2)*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

STATUS

approved