SEQUENCE TABLE OF RIGHT DIAGONALS

GENERATION OF RIGHT DIAGONALS FOR MAGIC SQUARE OF SQUARES INTRODUCTION

Square of Squares Tables

Andrew Bremner's article on squares of squares included the 3x3 square:

The numbers in the right diagonal as the tuple (97²,113²,127²) appears to have come out of the blue. But I will show that this sequence is part of a larger set of tuples having the same property, i.e. the first number in the tuple when added to a difference (Δ) gives the second square in the tuple and when this same (Δ) is added to the second square produces a third square. All these tuple sequences can be used as entries into the right diagonal of a magic square.

It was shown previously that these numbers are a part of a sequence of squares and this page is a continuation of that effort.

I will show from scratch, (i.e. from first principles) that these tuples (a²,b²,c²) whose sum a² + b² + c² − 3b² = 0 are generated from another set of tuples that obeys the equation a² + b² + c² − 3b² ≠ 0.

Generation of Sequences for the Production of Tables

The diagonal of a magic square consisting of square entries has the sum a² + b² + c² − 3b² = 0. If we treat the first entry a² as ±a² then the equation may be written as ±a² + c² = 2b² which on rearrangement gives c² = 2b² ± a² where a can be set initially to any number greater than 0. Let us set a = 1 and proceed as follows.

Table T_r consists the first seven tuples found using the formula c² = 2b² − 1. The second table T_i which will be used in the generation of complex squares, i.e., those with real and imaginary coefficients, consists of the first seven tuples found using the formula c² = 2b² + 1.

The desired c² is calculated by searching all b numbers between 1 and 100,000. However, it was found that the ratio of b_n+1/b_n or c_n+1/c_n converges on (1 + √2)² as the b's or c's get larger. This means that moving down each row on the table each integer value takes on the previous b_n or c_n multiplied by (1 + √2)², i.e., 5.8284271247...

In addition the entries in the columns may be ±1 giving eight combinations of (+ + +), (+ + −), (+ − +), (− + +), (+ − −), (− + −), (− − +) and (− − −) to generate eight Tuple tables. Each row in a tuple table will be used to generate secondary tables that will produce the requisite squares (a²,b²,c²) whose sum a² + b² + c² − 3b² = 0 . For example taking the tuple (1, 5, 7) will generate a sequence of tuples (a,b,c) which will obey the latter equation as is shown in square sequence table Part IA and from complex square sequence table Part IA using complex squares having five as well as six complex square entries. These squares may be the first instance of complex (real and imaginary coefficients).

If we look at the entries in the columns as a sequence of numbers, then the sequence of positive numbers from Table T_r columns 2 and column 3, c_n are the the known Sloane A001653 and Sloane A002315 numbers, respectively. For table T_i the sequences from the second and third columns are the known Sloane A001542 and Sloane A001541 numbers, respectively.

This concludes the introduction to the sequence table page.
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