OFFSET
1,2
COMMENTS
Every linear Diophantine equation with arbitrary integer coefficients may be reduced to this one.
The minimal nonnegative nonzero solutions form a generating system of the semigroup of all nonnegative solutions.
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.
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.
LINKS
M. Clausen, A. Fortenbacher, Efficient solution of linear Diophantine equations, J. Symbolic Comput. 8 (1989), 201-216.
D. V. Pasechnik, On computing Hilbert bases via the Elliott-MacMahon algorithm, Theor. Comp. Sc. 263 (2001), 37-46.
K. Pommerening, The indecomposable solutions of linear Diophantine equations
FORMULA
EXAMPLE
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).
PROG
(Python) See Pommerening link.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Klaus Pommerening, Dec 10 2017
STATUS
approved