A Bijective Proof of Richard Stanley's Observation that the sum of the cubes of the n-th row of Stern's Diatomic array equals 3 times 7 to the power n-1

TL;DR

This paper presents an elegant bijective proof for a formula discovered by Richard Stanley, showing that the sum of the cubes of the n-th row of Stern's diatomic array equals 3 times 7 to the power n-1, with the proof generated by a computer.

Contribution

It provides a novel bijective proof of Stanley's formula, discovered by a computer with minimal human guidance, demonstrating computer-assisted insight in combinatorics.

Findings

Bijective proof confirms Stanley's formula

Computer-assisted discovery of the bijection

Insight into combinatorial structures of Stern's array

Abstract

In a delightful article, Richard Stanley derived, algebraically, the surprisingly simple formula, 3 times 7 to the power n-1, for the sum of the cubes of the n-th row of Stern's diatomic array. In this note, we find an elegant bijective proof of this surprising fact, that explains it and gives insight. The novelty is that this gorgeous bijection was discovered by a computer (SBE), with minimal guidance by a human (DZ). This debunks the conventional wisdom, held by some human supremacists, that computers can only compute, but they can't give insight