We dive into A00126, exploring a nonlinear binomial-sum definition and its surprising connections to ternary numbers (digits 1 and 2 only, no zeros, with a digit-sum bound), Fibonacci spirals, and counting certain binary words. We also touch on the matrix-permanent interpretation and the various formulas (recurrences, closed form, generating functions) that let you compute the nth term. A compact tour of how one sequence threads together combinatorics, number theory, and linear algebra.

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