

A067997


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).


4



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
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OFFSET

0,2


COMMENTS

The U.S.A. issued the following unusual denomination coins during the 18th and 19th centuries: 1/2cent pieces, 17931857; 2cent pieces, 18641873; 3cent pieces, 18511889; and 20cent pieces, 18751878. This sequence is also the number of ways of making change for n cents using coins of 1 (two types, say, old pre1858 "large cents" and 1856topresent "small cents"), 2, 3, 5, 10, 20, 25, 50, 100 cents. For present purposes, one of the two types of 1cent piece is actually taken to be two 1/2cent pieces.


REFERENCES

R. S. Yeoman, A Guide Book of United States Coins, Ed. Kenneth Bressett, 53rd Edition (2000). New York: St. Martin's Press, 1999. pp. 7277, 9293, 104106, 135. (also known as The Official Red Book of United States Coins)


LINKS

Table of n, a(n) for n=0..46.
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.


FORMULA

G.f.: 1/((1x)^2*(1x^2)*(1x^3)*(1x^5)*(1x^10)*(1x^20)*(1x^25)*(1x^50)*(1x^100))


EXAMPLE

a(2)=4 because change can be made for 2 cents in these 4 ways: (1) 4 1/2cent coins, (2) 2 1/2cent, 1 1cent, (3) 2 1cent, (4) 1 2cent coin.


MATHEMATICA

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]


PROG

(PARI) a(n)=polcoeff(1/((1x)^2*(1x^2)*(1x^3)*(1x^5)*(1x^10)*(1x^20)*(1x^25)*(1x^50)*(1x^100)+x*O(x^n)), n)


CROSSREFS

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).
Sequence in context: A266901 A117649 A028291 * A175491 A034379 A007000
Adjacent sequences: A067994 A067995 A067996 * A067998 A067999 A068000


KEYWORD

easy,nonn


AUTHOR

Rick L. Shepherd, Feb 08 2002


STATUS

approved



