A296303 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.

1, 4, 13, 34, 99, 210, 559, 1164, 2531, 4940, 10735

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

Table of n, a(n) for n=1..11.

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

Lower and upper bounds (proved) are a(n) >= 2*A026905(n) for n >= 3 and a(n) <= A002894(n-1).

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

Cf. A096337, A026905, A002894.

Sequence in context: A357284 A208740 A127981 * A089453 A057159 A189588

Adjacent sequences: A296300 A296301 A296302 * A296304 A296305 A296306

Keyword

nonn,hard,more

Author

Klaus Pommerening, Dec 10 2017

Status

approved