[seqfan] n x n matrices with adjacent entries differing by +/- 1

Edwin Clark eclark at math.usf.edu
Mon Oct 27 23:41:58 CET 2008


On Mon, 27 Oct 2008, Benoît Jubin wrote:

> It would also be interesting to consider the entries of the matrix in
> Z/kZ (that is, 1 and k would also differ by 1). And also the same
> sequence for entries in Z or N, the upper-left term being 0.
>

Benoit:

The latter suggestion is actually more in the spirit of the problem that 
made me think of these sequences: Namely the Problem of the Month for 
mathematics students here at the University of South Florida:

A number of coins are placed on each square of a checkerboard such tha the 
amounts on every two squares having a common side differ by one cent. 
Give that the amount on one spare is 3 cents and another is 17 cents, find 
the total amount of money on both diagonals. [Don't post an answer since 
the problem is not due yet. :-)]

This problem was found by my colleague Mile Krajcevski in
a little booklet "Quantum Quandaries" (100 Brainteasers from QUANTUM , 
the magazine of math and science) , published in 1996 by the National
Science Teachers Association . The author of this problem is V.Proizvolov

Many similar questions can be raised. No need to restrict to nxn matrices 
try for n x m matrices or indeed replace the n x m grid by any graph.

I looked particularly at several n x 1 cases for various sets of entries 
and found some essentially identical sequences, but not with the same 
definition.

This seems a rather natural question. One can also look at the case where 
all entries differ by +/- q for different values of q. There is no end to 
it.

--Edwin



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