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%I #7 Oct 24 2019 14:33:28
%S 4,10,20,34,52,73,96,120,144,168,192,216,240,264,288,312,336,360,384,
%T 408,432,456,480,504,528,552,576,600,624,648,672,696,720,744,768,792,
%U 816,840,864,888,912,936,960,984,1008,1032,1056,1080,1104,1128,1152
%N Number of different values of exactly n standard American coins (pennies, nickels, dimes and quarters).
%H Andrew Howroyd, <a href="/A119651/b119651.txt">Table of n, a(n) for n = 1..250</a>
%F Conjectures from _Colin Barker_, Oct 24 2019: (Start)
%F G.f.: x*(4 + 2*x + 4*x^2 + 4*x^3 + 4*x^4 + 3*x^5 + 2*x^6 + x^7) / (1 - x)^2.
%F a(n) = 2*a(n-1) - a(n-2) for n>2.
%F a(n) = 24*(n-3) for n>6.
%F (End)
%e If you have 4 types of objects (coin denominations) you can have 35 different sets of 4. Out of these 35 only two have the same value: you can make 40 cents out of 4 dimes or out of a quarter and 3 nickels. Thus a(4) = 34.
%o (PARI) a(n) = {#select(k->k>0, Vec(polcoef(1/(1 - x*(y + y^5 + y^10 + y^25)) + O(x*x^n), n)))} \\ _Andrew Howroyd_, Oct 24 2019
%K nonn
%O 1,1
%A _Tanya Khovanova_, Jul 28 2006
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