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)
For n>0, the number of elements in every complete multigraph wherein the number of parallel edges between any two vertices equals the total number of vertices in the graph. Edge count follows A006002. - J.S. Seneschal, Mar 14 2026
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 entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: (1 + 2*x^2) / (1 - x)^4.
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
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
a(n) = n + A006002(n) + 1. - J.S. Seneschal, Mar 14 2026
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved