OFFSET
1,2
COMMENTS
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
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).
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
EXAMPLE
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).
MAPLE
A071816 := proc(n) n*(11*n^4+5*n^2+4)/20 ; end proc: # R. J. Mathar, Sep 04 2011
PROG
(Magma) [n*(11*n^4+5*n^2+4)/20: n in [1..30]]; // Vincenzo Librandi, Sep 05 2011
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Graeme McRae, Jun 07 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
STATUS
approved