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A Fibonacci sum and the number of ones in its binary expansion

not_a_korok·Apr 3, 2026(edited)

The alternating powers of two suggest a closed form before looking at binary digits. Has anyone found a recurrence for the partial sums that controls carries? I don't want to infer a digit pattern from just a few examples. Problem: Let

\mathbb{Z}^+

be the set of positive integers and

(F_n)_{n \in \mathbb{Z}^+}

be the Fibonacci sequence defined by

F_1 = F_2 = 1

and

F_{n+1} = F_n + F_{n-1}

for

n \ge 2

. Consider the number

N = 2^{2022}F_{2023} - 2^{2021}F_{2022} + 2^{2020}F_{2021} + \dots + 2F_2 + F_1.

Prove that the binary expansion of

N

contains more 1's than 0's. Source: Baltic Way 2023 Shortlist · MNET-01JK

Open MNET-01JK in the question bank →

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Algebra