This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339334 #17 Aug 17 2023 12:54:39 %S A339334 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 A339334 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 A339334 11,10,66,30,22,19,17,16,15,14,13,12,11 %N A339334 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 A339334 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 A339334 Pontus von Brömssen, <a href="/A339334/b339334.txt">Rows n = 1..40, flattened</a> %H A339334 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 A339334 Jeffrey Shallit, <a href="https://cs.uwaterloo.ca/~shallit/Papers/change2.pdf">What this country needs is an 18¢ piece</a>. %H A339334 Wikipedia, <a href="https://en.wikipedia.org/wiki/Change-making_problem">Change-making problem</a>. %F A339334 T(n,k) = A339333(n,k) for all k when 1 <= n <= 7 or n = 10. %F A339334 T(n,k) = A339333(n,k) for all n when k = 1 or k = 2. %F A339334 T(n,k) >= A339333(n,k). %F A339334 T(n,k) >= 2n - k, with equality if and only if n <= A014616(k). %e A339334 Triangle begins: %e A339334 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 %e A339334 ---|------------------------------------- %e A339334 1 | 1 %e A339334 2 | 3 2 %e A339334 3 | 6 4 3 %e A339334 4 | 10 6 5 4 %e A339334 5 | 15 9 7 6 5 %e A339334 6 | 21 11 9 8 7 6 %e A339334 7 | 28 14 11 10 9 8 7 %e A339334 8 | 36 18 14 12 11 10 9 8 %e A339334 9 | 45 21 17 14 13 12 11 10 9 %e A339334 10 | 55 25 19 16 15 14 13 12 11 10 %e A339334 11 | 66 30 22 19 17 16 15 14 13 12 11 %e A339334 12 | 78 33 25 22 19 18 17 16 15 14 13 12 %e A339334 For n = 8, one of the optimal greedy 3-coin systems is (1,2,4), with the representations %e A339334 1 = 1 %e A339334 2 = 2 %e A339334 3 = 2 + 1 %e A339334 4 = 4 %e A339334 5 = 4 + 1 %e A339334 6 = 4 + 2 %e A339334 7 = 4 + 2 + 1 %e A339334 8 = 4 + 4 %e A339334 with a total of 14 = T(8,3) terms. %e A339334 Shallit (2003) shows that T(99,k) is 4950, 900, 526, 410, 346, 313, 286 for k = 1..7. %Y A339334 Cf. A014616, A339333. %K A339334 nonn,tabl %O A339334 1,2 %A A339334 _Pontus von Brömssen_, Nov 30 2020