|
%I #149 Oct 27 2023 22:00:43
%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 D. N. Arnold, R. S. Falk, and R. 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 S. Butler and P. 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 S. 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) = numbperm(n,3)/2, n >= 2 [where numbperm(n, k) = n!/(n-k)!]. - _Zerinvary Lajos_, Apr 26 2007
%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
%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.
%K nonn,nice,easy
%O 0,2
%A _Olivier Gérard_ and Ken Knowlton (kcknowlton(AT)aol.com)
|
