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A366013
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Irregular triangle read by rows where each row lists coin denominations which make amounts 1 to 99 using the smallest total number of coins.
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3
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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
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(C) See links.
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CROSSREFS
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KEYWORD
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nonn,tabf,fini
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AUTHOR
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STATUS
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approved
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