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A339333
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 k-coin system of denominations.
4
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, 18, 13, 12, 11, 10, 9, 8, 45, 21, 16, 14, 13, 12, 11, 10, 9, 55, 25, 19, 16, 15, 14, 13, 12, 11, 10, 66, 30, 22, 18, 17, 16, 15, 14, 13, 12, 11
OFFSET
1,2
COMMENTS
T(n,k) <= A339334(n,k).
T(n,k) >= 2n - k, with equality if and only if n <= A001212(k).
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.
FORMULA
T(n,1) = A000217(n).
It appears that T(n,2) - T(n-1,2) = A322832(n).
T(n,k) = A339334(n,k) for all k when 1 <= n <= 7 or n = 10.
T(n,k) = A339334(n,k) for all n when k = 1 or k = 2.
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 13 12 11 10 9 8
9 | 45 21 16 14 13 12 11 10 9
10 | 55 25 19 16 15 14 13 12 11 10
11 | 66 30 22 18 17 16 15 14 13 12 11
12 | 78 33 24 20 19 18 17 16 15 14 13 12
For n = 8, there is a unique optimal 3-coin system (1,3,4), with the representations
1 = 1
2 = 1 + 1
3 = 3
4 = 4
5 = 4 + 1
6 = 3 + 3
7 = 4 + 3
8 = 4 + 4
with a total of 13 = T(8,3) terms.
Shallit (2003) shows that T(99,k) is 4950, 900, 515, 389, 329, 292, 265 for k = 1..7.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved