OFFSET
1,2
COMMENTS
An optimal greedy k-coin system of denominations for amounts 1..n is a set of k coin denominations such that the sum of the number of coins needed for each of the amounts 1, ..., n is as small as possible when the coins are chosen greedily, i.e., the largest coin value less than or equal to the remaining amount is always chosen.
LINKS
Pontus von Brömssen, Rows n = 1..40, flattened
Jeffrey Shallit, What this country needs is an 18¢ piece, The Mathematical Intelligencer 25 (2003), issue 2, 20-23.
Jeffrey Shallit, What this country needs is an 18¢ piece.
Wikipedia, Change-making problem.
FORMULA
EXAMPLE
Triangle begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12
---|-------------------------------------
1 | 1
2 | 3 2
3 | 6 4 3
4 | 10 6 5 4
5 | 15 9 7 6 5
6 | 21 11 9 8 7 6
7 | 28 14 11 10 9 8 7
8 | 36 18 14 12 11 10 9 8
9 | 45 21 17 14 13 12 11 10 9
10 | 55 25 19 16 15 14 13 12 11 10
11 | 66 30 22 19 17 16 15 14 13 12 11
12 | 78 33 25 22 19 18 17 16 15 14 13 12
For n = 8, one of the optimal greedy 3-coin systems is (1,2,4), with the representations
1 = 1
2 = 2
3 = 2 + 1
4 = 4
5 = 4 + 1
6 = 4 + 2
7 = 4 + 2 + 1
8 = 4 + 4
with a total of 14 = T(8,3) terms.
Shallit (2003) shows that T(99,k) is 4950, 900, 526, 410, 346, 313, 286 for k = 1..7.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Pontus von Brömssen, Nov 30 2020
STATUS
approved