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
Examples
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).
Links
- 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).
Programs
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Magma
[n*(11*n^4+5*n^2+4)/20: n in [1..30]]; // Vincenzo Librandi, Sep 05 2011
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Maple
A071816 := proc(n) n*(11*n^4+5*n^2+4)/20 ; end proc: # R. J. Mathar, Sep 04 2011
Formula
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
Extensions
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
Comments