A008528 -id:A008528 - cp's OEIS frontend

A071816 Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.

1, 20, 141, 580, 1751, 4332, 9331, 18152, 32661, 55252, 88913, 137292, 204763, 296492, 418503, 577744, 782153, 1040724, 1363573, 1762004, 2248575, 2837164, 3543035, 4382904, 5375005, 6539156, 7896825, 9471196, 11287235, 13371756

Offset: 1

Graeme McRae, Jun 07 2002

Number of 6-digit numbers in base n (with leading zeros allowed) such that the sum of the first three digits equals the sum of the last three digits.

a(n) = largest coefficient of (1+...+x^(n-1))^6. - R. H. Hardin, Jul 23 2009

			For n = 2 there are 20 ordered solutions (x,y,z,u,v,w) to x+y+z = u+v+w: (0,0,0,0,0,0), (0,0,1,0,0,1), (0,0,1,0,1,0), (0,0,1,1,0,0), (0,1,0,0,0,1), (0,1,0,0,1,0), (0,1,0,1,0,0), (0,1,1,0,1,1), (0,1,1,1,0,1), (0,1,1,1,1,0), (1,0,0,0,0,1), (1,0,0,0,1,0), (1,0,0,1,0,0), (1,0,1,0,1,1), (1,0,1,1,0,1), (1,0,1,1,1,0), (1,1,0,0,1,1), (1,1,0,1,0,1), (1,1,0,1,1,0), (1,1,1,1,1,1).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. B. Nathanson, Growth polynomials for additive quadruples and (h, k)-tuples, arXiv preprint arXiv:1305.7172 [math.NT], 2013.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008528, A071817, A077042.

First differences are in A070302.

[n*(11*n^4+5*n^2+4)/20: n in [1..30]]; // Vincenzo Librandi, Sep 05 2011

A071816 := proc(n) n*(11*n^4+5*n^2+4)/20 ; end proc: # R. J. Mathar, Sep 04 2011

The sum of the squares of the number of different 3-digit numbers that add up to k (summed over all possible k's) - cf. A071817.
a(n) = A077042(n,6).
a(n) = n*(11*n^4+5*n^2+4)/20. Recurrence: a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). G.f.: x*(1+14*x+36*x^2+14*x^3+x^4)/(1-x)^6. - Vladeta Jovovic, Jun 09 2002

New definition from Vladeta Jovovic, Jun 09 2002
Comment revised by Franklin T. Adams-Watters, Jul 27 2009
Edited by N. J. A. Sloane, Jul 28 2009

A070302 Number of 3 X 3 X 3 magic cubes with sum 3n.

Original entry on oeis.org

1, 19, 121, 439, 1171, 2581, 4999, 8821, 14509, 22591, 33661, 48379, 67471, 91729, 122011, 159241, 204409, 258571, 322849, 398431, 486571, 588589, 705871, 839869, 992101, 1164151, 1357669, 1574371, 1816039, 2084521, 2381731, 2709649

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 10 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Ahmed, J. De Loera, R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares, arXvi:0201108 [math.CO], 2002.
Maya Ahmed, Jesús De Loera and Raymond Hemmecke, Polyhedral cones of magic cubes and squares, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 25-41.
J. A. De Loera, D. C. Haws and M. Koppe, Ehrhart Polynomials of Matroid Polytopes and Polymatroids, arXiv:0710.4346 [math.CO], 2007; Discrete Comput. Geom., 42 (2009), 670-702.
D. C. Haws, Matroids [Broken link, Oct 30 2017]
D. C. Haws, Matroids [Copy on website of Matthias Koeppe]
D. C. Haws, Matroids [Cached copy, pdf file only]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

First differences are in A008528. Cf. A111085.

List([1..40],n->25*n^2/4-7*n/2-11*n^3/2+11*n^4/4+1); # Muniru A Asiru, Apr 30 2018

[25*n^2/4 -7*n/2 -11*n^3/2 +11*n^4/4+1: n in [1..40]]; // Vincenzo Librandi, Sep 05 2011

seq(25*n^2/4-7*n/2-11*n^3/2+11*n^4/4+1,n=1..40); # Muniru A Asiru, Apr 30 2018

Select[ CoefficientList[ Series[ (x^12 + 14x^9 + 36x^6 + 14x^3 + 1) / (1 - x^3)^5, {x, 0, 105}], x], # > 0 & ]
(* Second program: *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 19, 121, 439, 1171}, 32] (* Jean-François Alcover, Jan 07 2019 *)

for(n=1,30, print1(25*n^2/4 -7*n/2 -11*n^3/2 +11*n^4/4+1, ", ")) \\ G. C. Greubel, Apr 29 2018

G.f.: x*(x^4 + 14x^3 + 36x^2 + 14x + 1)/(1 - x)^5. [corrected by R. J. Mathar, Jan 26 2010]
a(n) = 25*n^2/4 - 7*n/2 - 11*n^3/2 + 11*n^4/4 + 1. - R. J. Mathar, Sep 04 2011
Sum_{n>=1} 1/a(n) = 2*Pi*(sqrt(17 + 4*sqrt(5)) * tanh(sqrt(17/44 - sqrt(5)/11)*Pi) - sqrt(17 - 4*sqrt(5))*tanh(sqrt(17/44 + sqrt(5)/11)*Pi)) / sqrt(95). - Vaclav Kotesovec, May 01 2018

Edited by Robert G. Wilson v, May 13 2002

cp's OEIS Frontend

A071816 Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Formula

Extensions