%I #17 Aug 17 2023 12:54:39

%S 1,3,2,6,4,3,10,6,5,4,15,9,7,6,5,21,11,9,8,7,6,28,14,11,10,9,8,7,36,

%T 18,14,12,11,10,9,8,45,21,17,14,13,12,11,10,9,55,25,19,16,15,14,13,12,

%U 11,10,66,30,22,19,17,16,15,14,13,12,11

%N Triangle read by rows, 1 <= k <= n: T(n,k) is the sum of the minimal number of coins needed for amounts 1..n with an optimal greedy k-coin system of denominations.

%C 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.

%H Pontus von Brömssen, <a href="/A339334/b339334.txt">Rows n = 1..40, flattened</a>

%H Jeffrey Shallit, <a href="http://dx.doi.org/10.1007/BF02984830">What this country needs is an 18¢ piece</a>, The Mathematical Intelligencer 25 (2003), issue 2, 20-23.

%H Jeffrey Shallit, <a href="https://cs.uwaterloo.ca/~shallit/Papers/change2.pdf">What this country needs is an 18¢ piece</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Change-making_problem">Change-making problem</a>.

%F T(n,k) = A339333(n,k) for all k when 1 <= n <= 7 or n = 10.

%F T(n,k) = A339333(n,k) for all n when k = 1 or k = 2.

%F T(n,k) >= A339333(n,k).

%F T(n,k) >= 2n - k, with equality if and only if n <= A014616(k).

%e Triangle begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12

%e ---|-------------------------------------

%e 1 | 1

%e 2 | 3 2

%e 3 | 6 4 3

%e 4 | 10 6 5 4

%e 5 | 15 9 7 6 5

%e 6 | 21 11 9 8 7 6

%e 7 | 28 14 11 10 9 8 7

%e 8 | 36 18 14 12 11 10 9 8

%e 9 | 45 21 17 14 13 12 11 10 9

%e 10 | 55 25 19 16 15 14 13 12 11 10

%e 11 | 66 30 22 19 17 16 15 14 13 12 11

%e 12 | 78 33 25 22 19 18 17 16 15 14 13 12

%e For n = 8, one of the optimal greedy 3-coin systems is (1,2,4), with the representations

%e 1 = 1

%e 2 = 2

%e 3 = 2 + 1

%e 4 = 4

%e 5 = 4 + 1

%e 6 = 4 + 2

%e 7 = 4 + 2 + 1

%e 8 = 4 + 4

%e with a total of 14 = T(8,3) terms.

%e Shallit (2003) shows that T(99,k) is 4950, 900, 526, 410, 346, 313, 286 for k = 1..7.

%Y Cf. A014616, A339333.

%K nonn,tabl

%O 1,2

%A _Pontus von Brömssen_, Nov 30 2020