A006000 a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2*x^2) / (1 - x)^4.

1, 4, 12, 28, 55, 96, 154, 232, 333, 460, 616, 804, 1027, 1288, 1590, 1936, 2329, 2772, 3268, 3820, 4431, 5104, 5842, 6648, 7525, 8476, 9504, 10612, 11803, 13080, 14446, 15904, 17457, 19108, 20860, 22716, 24679, 26752, 28938, 31240, 33661, 36204, 38872, 41668, 44595, 47656, 50854, 54192

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

Enumerates certain paraffins.

a(n) is the (n+1)st (n+3)-gonal number. - Floor van Lamoen, Oct 20 2001

Sum of n terms of an arithmetic progression with the first term 1 and the common difference n: a(1)=1, a(2) = 1+3, a(3) = 1+4+7, a(4) = 1+5+9+13, etc. - Amarnath Murthy, Mar 25 2004

This is identical to: first triangular number A000217, 2nd square number A000290, 3rd pentagonal number A000326, 4th hexagonal number A000384, 5th heptagonal number A000566, 6th octagonal number A000567, ..., (n+1)-th (n+3)-gonal number = main diagonal of rectangular array T(n,k) of polygonal numbers, by diagonals, referred to in A086271. - Jonathan Vos Post, Dec 19 2007

Also (n + 1)! times the determinant of the n X n matrix given by m(i,j) = (i+1)/i if i=j and otherwise 1. For example, (6 + 1)!*Det[{{2,1,1,1,1,1}, {1,3/2,1,1,1,1},{1,1,4/3,1,1,1}, {1,1,1,5/4,1,1}, {1,1,1,1,6/5,1}, {1,1,1,1,1,7/6}}] = 154 = a(6). - John M. Campbell, May 20 2011

a(n-1) = N_2(n), n>=1, is the number of 2-faces of n planes in generic position in three-dimensional space. See comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - Wolfdieter Lang, May 27 2011

For n>2, a(n) is 2 * (average cycle weight of primitive Hamiltonian cycles on a simply weighted K_n) (see link). - Jon Perry, Nov 23 2014

a(n) is the partial sums of A104249. - J. M. Bergot, Dec 28 2014

Sum of the numbers in the 1st column of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

From Enrique Navarrete, Mar 27 2023: (Start)

a(n) is the number of ordered set partitions of an (n+1)-set into 2 sets such that the first set has 0, 1, or 2 elements, the second set has no restrictions, and we choose an element from the second set. For n=4, the a(4) = 55 set partitions of [5] are the following (where the element selected from the second set is in parentheses):

{ }, {(1), 2, 3, 4, 5} (5 of these);

{1}, {(2), 3, 4, 5} (20 of these);

{1, 2}, {(3), 4, 5} (30 of these). (End)

References

• V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_2.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• William A. Tedeschi, Table of n, a(n) for n = 0..10000
• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
• S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
• P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-382. [See p. 301].
• Jon Perry, Weighted Hamiltonian Cycles
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Eric Weisstein's World of Mathematics, Polygonal Number
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000124.

Programs

Maple

A006000:=(1+2*z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

Mathematica

a[n_]:=(n^3-n^2)/2+n; Table[a[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
CoefficientList[Series[(1 + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

Python

def a(n):
    return n + (n**3 - n**2)//2 # William A. Tedeschi, Aug 22 2010

Formula

a(n) = Sum_{j=1..n+1} (binomial(0,0*j) + binomial(n+1,2)). - Zerinvary Lajos, Jul 25 2006

a(n-1) = n + (n^3 - n^2)/2 = n + n*T(n-1) where T(n-1) is a triangular number, n >= 1. - William A. Tedeschi, Aug 22 2010

a(n) = A002817(n)*4/n for n > 0. - Jon Perry, Nov 21 2014

E.g.f.: (1 + x)*(2 + 4*x + x^2)*exp(x)/2. - Robert Israel, Nov 24 2014

a(n) = A057145(n+3,n+1). - R. J. Mathar, Jul 28 2016

a(n) = A000124(n) * (n+1). - Alois P. Heinz, Aug 31 2023