0,2

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.

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)

Ron Guth, Your Online Reference For U.S. Coins

Mitch Hight, United States Coinage History

Clint Leland, U.S. Coins Quantities Minted

Index entries for sequences related to making change.

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))

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.

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]

(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)

Cf. A067996, A067995, A001314 (two kinds of nickels), A028291 (analogue for 1/2, 1, 2, 3, 5 only, or 1(two types), 2, 3, 5 only).

Sequence in context: A073471 A117649 A028291 this_sequence A034379 A007000 A073472

Adjacent sequences: A067994 A067995 A067996 this_sequence A067998 A067999 A068000

easy,nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 08 2002