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Solving Frobenius Equations and Computing Frobenius Numbers

A Frobenius equation is an equation of the form
where a1, ..., an are positive integers, m is an integer, and the coordinates x1, ..., xn of solutions are required to be non-negative integers.
The Frobenius number of a1, ..., an is the largest integer m for which the Frobenius equation a1x1+...+anxn= m has no solutions.
FrobeniusSolve[{a1,...,an},b]give a list of all solutions of the Frobenius equation a1x1+...+anxn=b
FrobeniusSolve[{a1,...,an},b,m]give m solutions of the Frobenius equation a1x1+...+anxn=b; if less than m solutions exist, give all solutions
FrobeniusNumber[{a1,...,an}]give the Frobenius number of a1, ..., an

Functions for solving Frobenius equations and computing Frobenius numbers.

This gives all solutions of the Frobenius equation 12x+16y+20z+27tEqual123.
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This gives one solution of the Frobenius equation 12x+16y+20z+27tEqual123.
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Here is the Frobenius number of {12, 16, 20, 27}, that is, the largest m for which the Frobenius equation 12x+16y+20z+27tEqualm has no solutions.
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This shows that indeed, the Frobenius equation 12x+16y+20z+27tEqual89 has no solutions.
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Here are all the ways of making 42 cents change using 1, 5, 10, and 25 cent coins.
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Using 24, 29, 31, 34, 37, and 39 cent stamps, you can pay arbitrary postage of more than 88 cents.
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