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%I #25 Dec 14 2017 03:48:23
%S 1,4,13,34,99,210,559,1164,2531,4940,10735
%N Number of minimal nonnegative nonzero solutions of the linear Diophantine equation x_1 + 2*x_2 + ... + n*x_n = y_1 + 2*y_2 + ... + n*y_n.
%C Every linear Diophantine equation with arbitrary integer coefficients may be reduced to this one.
%C The minimal nonnegative nonzero solutions form a generating system of the semigroup of all nonnegative solutions.
%C The asymptotic behavior of a(n) is unknown, it is somewhere between a*exp(b*sqrt(n))/(sqrt(n)) and c*exp(d*n)/n with positive real numbers a,b,c,d.
%C A096337 contains the number of minimal nonnegative nonzero solutions of the linear congruence x_1 + 2 x_2 + ... + (n-1) x_{n-1} == 0 (mod n). There is an obvious relation with a(n) since every solution (x_1, ..., x_{n-1}) of the linear congruence yields a solution (x_1, ..., x_{n-1}; 0, 0, ..., 0, k) of the linear Diophantine equation.
%H M. Clausen, A. Fortenbacher, <a href="https://doi.org/10.1016/S0747-7171(89)80025-2">Efficient solution of linear Diophantine equations</a>, J. Symbolic Comput. 8 (1989), 201-216.
%H D. V. Pasechnik, <a href="https://doi.org/10.1016/S0304-3975(00)00229-2">On computing Hilbert bases via the Elliott-MacMahon algorithm</a>, Theor. Comp. Sc. 263 (2001), 37-46.
%H K. Pommerening, <a href="http://www.staff.uni-mainz.de/pommeren/MathMisc/LinDio.pdf">The indecomposable solutions of linear Diophantine equations</a>
%F Lower and upper bounds (proved) are a(n) >= 2*A026905(n) for n >= 3 and a(n) <= A002894(n-1).
%e The 13 minimal solutions for n=3 are (x-coordinates followed by y-coordinates): (0,0,1;0,0,1), (0,0,1;1,1,0), (0,0,1;3,0,0), (0,0,2;0,3,0), (0,1,0;0,1,0), (0,1,0;2,0,0), (0,2,0;1,0,1), (0,3,0;0,0,2), (1,0,0;1,0,0), (1,0,1;0,2,0), (1,1,0;0,0,1), (2,0,0;0,1,0), (3,0,0;0,0,1).
%o (Python) See Pommerening link.
%Y Cf. A096337, A026905, A002894.
%K nonn,hard,more
%O 1,2
%A _Klaus Pommerening_, Dec 10 2017
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