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A067997
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Number of (unordered) ways of making change for n cents using coins of 1/2, 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage denominations up to 100 cents).
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5
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1, 2, 4, 7, 11, 17, 25, 35, 48, 64, 85, 110, 141, 178, 222, 275, 337, 409, 493, 589, 702, 830, 977, 1144, 1333, 1549, 1792, 2065, 2372, 2714, 3100, 3528, 4005, 4534, 5119, 5769, 6485, 7273, 8140, 9089, 10135, 11276, 12524, 13885, 15366, 16983, 18738
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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The U.S.A. issued the following unusual denomination coins during the 18th and 19th centuries: 1/2-cent pieces, 1793-1857; 2-cent pieces, 1864-1873; 3-cent pieces, 1851-1889; and 20-cent pieces, 1875-1878. This sequence is also the number of ways of making change for n cents using coins of 1 (two types, say, old pre-1858 "large cents" and 1856-to-present "small cents"), 2, 3, 5, 10, 20, 25, 50, 100 cents. For present purposes, one of the two types of 1-cent piece is actually taken to be two 1/2-cent pieces.
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REFERENCES
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R. S. Yeoman, A Guide Book of United States Coins, Ed. Kenneth Bressett, 53rd Edition (2000). New York: St. Martin's Press, 1999. pp. 72-77, 92-93, 104-106, 135. (also known as The Official Red Book of United States Coins)
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LINKS
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FORMULA
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G.f.: 1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^25)*(1-x^50)*(1-x^100))
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EXAMPLE
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a(2)=4 because change can be made for 2 cents in these 4 ways: (1) 4 1/2-cent coins, (2) 2 1/2-cent, 1 1-cent, (3) 2 1-cent, (4) 1 2-cent coin.
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MATHEMATICA
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CoefficientList[ Series[1 / ((1 - x)^2(1 - x^2)(1 - x^3)(1 - x^5)(1 - x^10)(1 - x^20)(1 - x^25)(1 - x^50)(1 - x^100)), {x, 0, 50} ], x]
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PROG
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(PARI) a(n)=polcoeff(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^25)*(1-x^50)*(1-x^100)+x*O(x^n)), n)
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CROSSREFS
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Cf. A067996, A067995, A001314 (two kinds of nickels), A028291 (analog for 1/2, 1, 2, 3, 5 only, or 1(two types), 2, 3, 5 only).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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