Sums payable by exactly n banknotes.

[David Wilson]

Let S(n) be the number of amounts achievable using n bills. You are 
interested in a(n) = #S(n). Don't forget a(0) = 1 (the single value 0 is 
achievable with 0 coins/bills).

Clearly, if gcd(denominatoins) = g > 1, we can divide all denominations by g 
without affecting the counts, so WLOG we can assume gcd(denominations) = 1.

At any rate, my guess is that for sufficient n, S(n) includes a large 
maximal middle interval M(n) = [x, y] in which all amounts are achievable, 
and head and tail sections H(n) = S(n) intersect [0, x-1] and T(n) = S(n) 
intersect [y+1, inf]. I suspect that for sufficient n,  |M(n)| = (hi-lo)n + 
c, where hi and lo are the high and low denominations, while the translated 
sets H(n)-x and T(n)-y become constant, or at least periodic in n. This 
suggests that in the general case,

    |S(n)| = ((hi-lo)/g) n + p(n)

for sufficient n, where p(n) is a periodic function of n.