A006000 a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2*x^2) / (1 - x)^4. (Formerly M3436)

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

0,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A000124.

Sequence in context: A112087 A166019 A184633 * A161216 A085622 A011940

Adjacent sequences: A005997 A005998 A005999 * A006001 A006002 A006003

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved