OEIS A00008: Number of ways to make change and Pick's Theorem

Explore how a simple daily task—making change—unfolds into rich math. OEIS A00008 counts the ways to make change with US coins, and plotting the nickels-and-dimes solutions yields a triangular lattice that leads to counting formulas. The theorems extend to general amounts (multiples of five and non-multiples), incorporate quarters and half-dollars, and even touch a surprising link to primes and semiprimes. The discussion then shifts to Pick's Theorem, showing how the area of lattice polygons can be computed from interior and boundary points. A window into how everyday problems hide elegant connections between combinatorics, number theory, and geometry.

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