OEIS A000115: Partitions into 1s, 2s, and 5s

In this episode we explore A000115, the denumerant counting the number of ways to write n as a nonnegative sum of 1, 2, and 5. We’ll uncover the surprisingly simple closed form A(n) = round((n+4)^2/20), discuss the generating function 1/((1 - x)(1 - x^2)(1 - x^5)), and interpret A000115 as the number of nonnegative solutions to x1 + 2x2 + 5x3 = n. We’ll connect the sequence to the classic coin-change problem, explore a basic recurrence, and note the symmetry A(n) = A(-n-8). Along the way we’ll situate A000115 within the broader OEIS network and touch on James Joseph Sylvester’s role in developing the theory of denumerants.

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