OEIS A000121: Representing integers as sums of Fibonacci numbers with a doubled 1

We explore A000121, the counting sequence for representing every positive integer as a sum of Fibonacci numbers when the number 1 may be used twice. From Zeckendorf-style variants and connections to A000119 (partitions into distinct parts) to the inverse Euler transform and Fibonacci coding, we uncover the rich structural ties behind this simple rule. We’ll touch on generating functions, asymptotics, and even surprising links to geometry and cryptography, highlighting why this tiny step beyond Zeckendorf opens a world of math.

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