A027480 - OEIS

%I #181 Jul 08 2024 10:37:35

%S 0,3,12,30,60,105,168,252,360,495,660,858,1092,1365,1680,2040,2448,

%T 2907,3420,3990,4620,5313,6072,6900,7800,8775,9828,10962,12180,13485,

%U 14880,16368,17952,19635,21420,23310,25308,27417,29640,31980,34440

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

%C 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

%C 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

%C 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

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

%C Alternate terms of A057587. - _Jeremy Gardiner_, Apr 10 2005

%C 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

%C 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

%C 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

%C From _Martin Licht_, Dec 04 2016: (Start)

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

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

%C 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.

%C 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)

%C 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

%H T. D. Noe, <a href="/A027480/b027480.txt">Table of n, a(n) for n = 0..1000</a>

%H Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, <a href="http://www-users.math.umn.edu/~arnold/papers/acta.pdf">Finite element exterior calculus, homological techniques, and applications</a>, Acta Numerica 15 (2006), 1-155.

%H Steve Butler and Pavel Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. 13 (2010), 10.4.4.

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 9.

%H Solomon Gartenhaus, <a href="https://arxiv.org/abs/math/0210275">Odd Order Pandiagonal Latin and Magic Cubes in Three and Four Dimensions</a>, arXiv:math/0210275 [math.CO], 2002.

%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F a(n) = n*binomial(2+n, 2). - _Zerinvary Lajos_, Jan 10 2006

%F a(n) = A007531(n+2)/2. - _Zerinvary Lajos_, Jul 17 2006

%F Starting with offset 1 = binomial transform of [3, 9, 9, 3, 0, 0, 0]. - _Gary W. Adamson_, Oct 25 2007

%F From _R. J. Mathar_, Apr 07 2009: (Start)

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

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

%F a(n) = Sum_{i=0..n} n*(n - i) + 2*i. - _Bruno Berselli_, Jan 13 2016

%F From _Ilya Gutkovskiy_, Aug 07 2016: (Start)

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

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

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

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

%F a(n-1) = binomial(n^2,2)/n for n > 0. - _Jonathan Sondow_, Jan 07 2018

%F 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

%e Row sums of n consecutive integers, starting at 0, seen as a triangle:

%e .

%e     0 |  0

%e     3 |  1  2

%e    12 |  3  4  5

%e    30 |  6  7  8  9

%e    60 | 10 11 12 13 14

%e   105 | 15 16 17 18 19 20

%p [seq(3*binomial(n+2,3),n=0..37)]; # _Zerinvary Lajos_, Nov 24 2006

%p a := n -> add((j+n)*(n+2)/3,j=0..n): seq(a(n),n=0..35); # _Zerinvary Lajos_, Dec 17 2006

%t Table[(m^3 - m)/2, {m, 36}] (* _Zerinvary Lajos_, Mar 21 2007 *)

%t LinearRecurrence[{4,-6,4,-1},{0,3,12,30},40] (* _Harvey P. Dale_, Oct 10 2012 *)

%t CoefficientList[Series[3 x / (x - 1)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 14 2014 *)

%t 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 *)

%o (PARI) a(n)=3*binomial(n+2,3) \\ _Charles R Greathouse IV_, May 23 2011

%o (Magma) [n*(n+1)*(n+2)/2: n in [0..40]]; // _Vincenzo Librandi_, Nov 14 2014

%o (Python) def a(n): return (n**3+3*n**2+2*n)//2 # __Torlach Rush_, Jun 16 2024

%Y 1/beta(n, 3) in A061928.

%Y Cf. A057587, A006003, A254407.

%Y A row of array in A129533.

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

%Y Similar sequences are listed in A316224.

%Y Cf. A056923, A281258.

%Y Third column of A003506.

%Y Cf. A000217, A002378, A006002.

%K nonn,nice,easy

%O 0,2

%A _Olivier Gérard_ and Ken Knowlton (kcknowlton(AT)aol.com)