A364607 -id:A364607 - cp's OEIS frontend

A208953 Amounts (in cents) of coins in denominations suggested by Shallit.

1, 5, 10, 18, 25, 50

Offset: 1

Jonathan Vos Post, May 31 2012

The following is quoted (with minor changes) from Alan Burdick's article: "Jeffrey Shallit analyzed the average handful of change, and devised a clever way to reduce its size. Getting rid of the 1-cent coin, a plot advocated by numerous antipennyists, would certainly help, he says. But Shallit's own scheme for reducing loose change involves the creation of an entirely new coin. What the United States needs, he says, is an 18-cent piece. Shallit reached this conclusion by a linear Diophantine equation. Shallit calculated that the average U.S. transaction produces 4.7 coins in change. If we got rid of the dime and replaced it with an 18-cent coin, the 'cost' of the average transaction would drop from 4.7 to 3.89 coins. A system of coins worth 1¢, 5¢, 18¢, and 29¢ would have the same effect. Should we wish to keep the dime and simply add a fifth denomination, the best coin to add would be 32¢, for an efficiency of 3.46. Even better, if we kept the dime and actually used the half-dollar, then added an 18-cent coin to that mix, we'd gain maximum efficiency: You'd get back a mere 3.18 coins per transaction."

Alan Burdick, The Physics of ... Pocket Change, Discover, October 2003 issue; published online October 1, 2003.
Thomas Young, Change the Dime not the Dollar!, preprint, 1995.

Cf. A212773, A212774.

See A364607 for another version. - N. J. A. Sloane, Aug 12 2023

A366013 Irregular triangle read by rows where each row lists coin denominations which make amounts 1 to 99 using the smallest total number of coins.

Original entry on oeis.org

1, 1, 10, 1, 11, 1, 12, 19, 1, 5, 18, 25, 1, 5, 18, 29, 1, 5, 16, 23, 33, 1, 4, 6, 21, 30, 37, 1, 5, 8, 20, 31, 33, 1, 4, 9, 11, 26, 38, 44, 1, 3, 8, 9, 20, 30, 44, 48, 1, 3, 4, 9, 16, 27, 37, 44, 49, 1, 3, 4, 10, 17, 25, 37, 43, 48, 1, 3, 4, 10, 18, 22, 31, 42, 47

Offset: 1

Kevin Ryde, Sep 28 2023

A row of length d makes amounts 1 to 99 using a total of A339333(99,d) coins, which is the minimum possible for d denominations.
Denominations within a row are in ascending order and rows are ordered by length and then lexicographically.
Each row starts with denomination 1 since 1 is the only way to make amount 1.
This is a finite sequence, ending with a row of all denominations 1 to 99 which make all amounts using a single coin each.
Amounts 1 to 99 are based on making change in a decimal currency which uses coins for 1 to 99 cents, and notes for whole dollar parts.
Minimizing the total number of coins minimizes the average number of coins given as change, assuming each of 1 to 99 are equally likely amounts to be given.

			Triangle begins:
      k=1   2   3   4   5   6
  n=1:  1
  n=2:  1, 10
  n=3:  1, 11
  n=4:  1, 12, 19
  n=5:  1,  5, 18, 25
  n=6:  1,  5, 18, 29
  n=7:  1,  5, 16, 23, 33
  n=8:  1,  4,  6, 21, 30, 37
  n=9:  1,  5,  8, 20, 31, 33
Rows n=5 and n=6 are of length d=4 and are the two sets of denominations which can make amounts 1 to 99 using the minimum total of A339333(99,4) = 389 coins.

Kevin Ryde, Table of n, a(n) for rows 1..600 (lengths d=1..16 and some d=17), flattened
Kevin Ryde, C Code
Jeffrey Shallit, What This Country Needs is an 18¢ Piece, The Mathematical Intelligencer, 25-2, pages 20-23, 2003, figure 1 rows to d=7, and also author's copy, 2002.
Thomas Young, Change the Dime, not the Dollar, 1995, first set of denominations d=4 (see A364607).

Cf. A339333, A364607 (row n=5).

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A208953 Amounts (in cents) of coins in denominations suggested by Shallit.

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