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%I #30 Jan 15 2024 11:02:03
%S 1,20,141,580,1751,4332,9331,18152,32661,55252,88913,137292,204763,
%T 296492,418503,577744,782153,1040724,1363573,1762004,2248575,2837164,
%U 3543035,4382904,5375005,6539156,7896825,9471196,11287235,13371756
%N Number of ordered solutions to x+y+z = u+v+w, 0 <= x, y, z, u, v, w < n.
%C 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.
%C a(n) = largest coefficient of (1+...+x^(n-1))^6. - _R. H. Hardin_, Jul 23 2009
%H Vincenzo Librandi, <a href="/A071816/b071816.txt">Table of n, a(n) for n = 1..10000</a>
%H M. B. Nathanson, <a href="http://arxiv.org/abs/1305.7172">Growth polynomials for additive quadruples and (h, k)-tuples</a>, arXiv preprint arXiv:1305.7172 [math.NT], 2013.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F 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.
%F a(n) = A077042(n,6).
%F 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
%e 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).
%p A071816 := proc(n) n*(11*n^4+5*n^2+4)/20 ; end proc: # _R. J. Mathar_, Sep 04 2011
%o (Magma) [n*(11*n^4+5*n^2+4)/20: n in [1..30]]; // _Vincenzo Librandi_, Sep 05 2011
%Y Cf. A008528, A071817, A077042.
%Y First differences are in A070302.
%K nonn,base,easy
%O 1,2
%A _Graeme McRae_, Jun 07 2002
%E New definition from _Vladeta Jovovic_, Jun 09 2002
%E Comment revised by _Franklin T. Adams-Watters_, Jul 27 2009
%E Edited by _N. J. A. Sloane_, Jul 28 2009
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