OEIS A000064: Ways to Make Change with 1, 2, 5, and 10-Cent Coins

We explore A000064, the coin-change sequence that counts, for each n, the number of ways to make change using 1, 2, 5, and 10-cent coins. We unpack its combinatorial meaning as partitions into those four part sizes, examine the compact generating function 1/((1−x)(1−x^2)(1−x^5)(1−x^10)), and discuss the accompanying recurrence and the asymptotic relation a(n) ~ n^4/2400. Along the way we’ll see how dynamic programming and more advanced convolution techniques compute large terms, why the general change-making problem remains algorithmically rich (from greedy pitfalls to the idea of canonical coin systems), and how historical and combinatorial viewpoints—like Jor Arndt’s partition interpretation—shed light on this deceptively simple problem.

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