Extrema of a(n)/n^(log_phi 2) for the Fibbinary Numbers on Intervals Between Consecutive Fibonacci Numbers

Abstract

This note studies the Fibbinary numbers, OEIS A003714, through the map that transfers the digits of the Zeckendorf representation of an integer to the corresponding binary place values. Writing κ=log⁡φ2\kappa=\log_\varphi 2, it determines the exact maximum and minimum of a(n)/nκa(n)/n^\kappa on every interval between consecutive Fibonacci numbers. The maximizer is the left endpoint FkF_k, the minimizer is the right endpoint Fk+1−1F_{k+1}-1, and both are unique for k≥4k\ge4; explicit endpoint values give sharp finite inequalities throughout each interval. The resulting liminf and limsup are recovered as consequences of these finite extrema and agree, under monotone inversion, with the earlier counting-function constants proved by Sungkon Chang. The exact finite Fibonacci-interval theorem and its equality classification are the principal contribution.