SEQUENCE TABLE OF RIGHT DIAGONALS

GENERATION OF RIGHT DIAGONALS FOR MAGIC SQUARE OF SQUARES (Intro II)

Picture of a square

Square of Squares Tables

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

Bremmer's square
5824621272
942113222
972822742

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 + c23b2 = 0 are generated from another set of tuples that obeys the equation a2 + b2 + c23b20.

Generation of Sequences for the Production of Tables

The diagonal of a magic square consisting of square entries has the sum a2 + b2 + c23b2 = 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 Tr consists the first seven tuples found using the formula c2 = 2b2 − 2. The second table Ti 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 + 2.

The desired c2 is calculated by searching all b numbers between 1 and 100,000. However, it was found that the ratio of bn+1/bn or cn+1/cn 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 bn or cn 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 + c23b2 = 0 . For example taking the tuple (√2, 3, 4) will generate a sequence of tuples (a,b,c) which will obey the latter equation as is shown in square sequence table Part IA.

If we look at the entries in the columns as a sequence of numbers, then the sequence of positive numbers from Table Tr columns 2 and column 3, cn are the the known Sloane A001541 and Sloane A005319 numbers, respectively. For table Ti the sequences from the second and third columns are the known Sloane A002315 and Sloane A075870, respectively.

Table Tr
anbncn
±√2±1±0
±√2±3±4
±√2±17±24
±√2±99±140
±√2±577±816
±√2±3363±4756
±√2±19601±27720
 
Table Ti
anbncn
±√2 i±1±2
±√2 i±7±10
±√2 i±41±58
±√2 i±239±338
±√2 i±1393±1970
±√2 i±8119±11482
±√2 i±47321±390050

This concludes the introduction to the sequence table page.
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Copyright © 2011 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com