

A027480


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


53



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
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OFFSET

0,2


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 (asher.auel(AT)reed.edu), 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 00, 01, 02, 03, 11, 12, 13, 22, 23, 33.)  Gerard Schildberger, Jun 26 2003. See A129533 for generalization to narmed "dominoes".  N. J. A. Sloane, Jan 06 2016
Common sum in an (n+1) X (n+1) magic square with entries (0..n^21).
Alternate terms of A057587.  Jeremy Gardiner, Apr 10 2005
If Y is a 3subset of an nset X then, for n >= 5, a(n5) is the number of 4subsets of X which have exactly one element in common with Y. Also, if Y is a 3subset of an nset X then, for n >= 5, a(n5) is the number of (n5)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) = (1x)^2 * log(1/(1x)), 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(n1) is the dimension of the curlconforming 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 divergenceconforming 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


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta numerica 15 (2006), 1155.
S. Butler and P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4.
S. 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).


FORMULA

a(n) = a(n1) + A050534(n) = 3*A000292(n1) = A050534(n)  A050534(n1).
a(n) = n*binomial(2+n, 2).  Zerinvary Lajos, Jan 10 2006
a(n) = numbperm(n,3)/2, n >= 2 [where numbperm(n, k) = n!/(nk)!].  Zerinvary Lajos, Apr 26 2007
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/(x1)^4.
a(n) = 4*a(n1)  6*a(n2) + 4*a(n3)  a(n4). (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(n1) = binomial(n^2,2)/n for n > 0.  Jonathan Sondow, Jan 07 2018
For k > 1, Sum_{i=0..n^21} (k+i)^2 = (k*n + a(k1))^2 + A126275(k).  Charlie Marion, Apr 23 2021


EXAMPLE

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


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))/21], Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jun 02 2019 *)


PROG

(PARI) a(n)=3*binomial(n+2, 3) \\ Charles R Greathouse IV, May 23 2011
(Magma) [n*(n+1)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Nov 14 2014


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.
Sequence in context: A331080 A164013 A057671 * A135503 A048088 A064181
Adjacent sequences: A027477 A027478 A027479 * A027481 A027482 A027483


KEYWORD

nonn,nice,easy


AUTHOR

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


STATUS

approved



