By Shalosh B. Ekhad and Doron Zeilberger
First Written: July 27, 2021 ; This version: Aug. 4, 2021 .
[To appear in the Palestinian Journal of Mathematics]
A few years ago, Peter Larcombe discovered an amazing property regarding two by two matrices. For any such 2 times 2 matrix A, the ratios of the two anti-diagonal entries is the same for all powers of A. We discuss extensions to higher dimensions, and give a short bijective proof of Larcombe and Eric Fennessey's elegant extension to tri-diagonal matrices of arbitrary dimension. This article is accompanied by a Maple package.
Added Sept. 29, 2021: Read the insightful comments by Darij Grinberg.
Added Aug. 4, 2021: With Darij Grinberg's kind permission, this is now included in the actual article.
the input file yields the output file
the input file yields the output file
Note: This sequence was not (Aug. 8, 2021) in the OEIS. On the other hand the total number of such graphs is in the OEIS.
[It uses Polya-Redfield theory]
The input file yields the output file