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<sup>n-1</sup>

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^n-1
By Shalosh B. Ekhad and Doron Zeilberger
.pdf .ps .tex

Written: March 3, 2021
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org
In a delightful article, Richard Stanley derived, algebraically, the surprisingly simple formula, 3*7^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.

Maple package

BijectionStern.txt, a Maple package that discoveres and implements the bijection of the paper

Sample Input and Output Files for BijectionStern.txt

If you want to see how the bijection acts for all balanced 3 by n matrices for n from 1 to 5,
The input file gives you the output file .

Personal Journal of SBE and DZ
Doron Zeilberger's Home Page