A027480 a(n) = n*(n+1)*(n+2)/2.

0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Author

Olivier Gérard and Ken Knowlton (kcknowlton(AT)aol.com)

Keywords

• nonn
• nice
• easy

Comments

Write the integers in groups: 0; 1,2; 3,4,5; 6,7,8,9; ... and add the groups: a(n) = Sum_{j=0..n} (A000217(n)+j), row sums of the triangular view of A001477. - Asher Auel, Jan 06 2000

With offset = 2, a(n) is the number of edges of the line graph of the complete graph of order n, L(K_n). - Roberto E. Martinez II, Jan 07 2002

Also the total number of pips on a set of dominoes of type n. (A "3" domino set would have 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, 3-3.) - Gerard Schildberger, Jun 26 2003. See A129533 for generalization to n-armed "dominoes". - N. J. A. Sloane, Jan 06 2016

Common sum in an (n+1) X (n+1) magic square with entries (0..n^2-1).

Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005

If Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of 4-subsets of X which have exactly one element in common with Y. Also, if Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-5)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

These numbers, starting with 3, are the denominators of the power series f(x) = (1-x)^2 * log(1/(1-x)), if the numerators are kept at 1. This sequence of denominators starts at the term x^3/3. - Miklos Bona, Feb 18 2009

a(n) is the number of triples (w,x,y) having all terms in {0..n} and satisfying at least one of the inequalities x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012

From Martin Licht, Dec 04 2016: (Start)

Let b(n) = (n+1)(n+2)(n+3)/2 (the same sequence, but with a different offset). Then (see Arnold et al., 2006):

b(n) is the dimension of the Nédélec space of the second kind of polynomials of order n over a tetrahedron.

b(n-1) is the dimension of the curl-conforming Nédélec space of the first kind of polynomials of order n with tangential boundary conditions over a tetrahedron.

b(n) is the dimension of the divergence-conforming Nédélec space of the first kind of polynomials of order n with normal boundary conditions over a tetrahedron. (End)

After a(0), the digital root has period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9]. - Peter M. Chema, Jan 19 2017

Examples

			Row sums of n consecutive integers, starting at 0, seen as a triangle:
.
    0 |  0
    3 |  1  2
   12 |  3  4  5
   30 |  6  7  8  9
   60 | 10 11 12 13 14
  105 | 15 16 17 18 19 20
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006), 1-155.
• Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4.
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
• Solomon Gartenhaus, Odd Order Pandiagonal Latin and Magic Cubes in Three and Four Dimensions, arXiv:math/0210275 [math.CO], 2002.
• Index entries for sequences related to dominoes
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

1/beta(n, 3) in A061928.

Cf. A057587, A006003, A254407.

A row of array in A129533.

Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.

Similar sequences are listed in A316224.

Cf. A056923, A281258.

Third column of A003506.

Cf. A000217, A002378, A006002.

A bisection of A330298.

Programs

Magma

[n*(n+1)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Nov 14 2014

Maple

[seq(3*binomial(n+2,3),n=0..37)]; # Zerinvary Lajos, Nov 24 2006
a := n -> add((j+n)*(n+2)/3,j=0..n): seq(a(n),n=0..35); # Zerinvary Lajos, Dec 17 2006

Mathematica

Table[(m^3 - m)/2, {m, 36}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{4,-6,4,-1},{0,3,12,30},40] (* Harvey P. Dale, Oct 10 2012 *)
CoefficientList[Series[3 x / (x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
With[{nn=50},Total/@TakeList[Range[0,(nn(nn+1))/2-1],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jun 02 2019 *)

PARI

a(n)=3*binomial(n+2,3) \\ Charles R Greathouse IV, May 23 2011

Python

def a(n): return (n**3+3*n**2+2*n)//2 # _Torlach Rush, Jun 16 2024

Formula

a(n) = a(n-1) + A050534(n) = 3*A000292(n-1) = A050534(n) - A050534(n-1).

a(n) = n*binomial(2+n, 2). - Zerinvary Lajos, Jan 10 2006

a(n) = A007531(n+2)/2. - Zerinvary Lajos, Jul 17 2006

Starting with offset 1 = binomial transform of [3, 9, 9, 3, 0, 0, 0]. - Gary W. Adamson, Oct 25 2007

From R. J. Mathar, Apr 07 2009: (Start)

G.f.: 3*x/(x-1)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

a(n) = Sum_{i=0..n} n*(n - i) + 2*i. - Bruno Berselli, Jan 13 2016

From Ilya Gutkovskiy, Aug 07 2016: (Start)

E.g.f.: x*(6 + 6*x + x^2)*exp(x)/2.

a(n) = Sum_{k=0..n} A045943(k).

Sum_{n>=1} 1/a(n) = 1/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (8*log(2) - 5)/2 = 0.2725887222397812... = A016639/10. (End)

a(n-1) = binomial(n^2,2)/n for n > 0. - Jonathan Sondow, Jan 07 2018

For k > 1, Sum_{i=0..n^2-1} (k+i)^2 = (k*n + a(k-1))^2 + A126275(k). - Charlie Marion, Apr 23 2021