NEW SEQUENCE OF SQUARES

GENERATION OF PARTIAL MAGIC SQUARE OF SQUARES

Square of Square Sequences

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 sequence of numbers which can be broken down into 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.

These sequnces will be shown to have special properties where the diffence between numbers generates other sequences.

Construction of two Sequences

1. The first sequence constructed generates a non-square Sequence I.
2. We begin using equation ½(n² - 2n - 1) where n is initially 3. This gives the number 1 in Sequence I.
3. Applying equation ½(n² + 1) next gives us the number 5.
4. Then applying equation ½(n² + 2n - 1) next gives us the number 7.
5. This completes one cycle. Incrementing n by two gives us 5 and the next number to substitute into the first equation to repeat the cycle.
6. Since the number generated using n₁ for n in ½(n² + 2n - 1) is identical to ( n₁ +2 ) for n in ½(n² - 2n - 1) the second identical number is not used in the sequence.

This can be generalized tothe following sequence having the Sloane Number A178218: which may be found in the catalogue of numbers:

1. The sequence can be generated by an alternative general method The Generation of New Sequences (Part G) discovered in 9/2012 and stored on this website.
2. The sequence generated goes up to the number 577 and is shown below in white. (To generate a larger sequence see below).
3. The second sequence in green and tan shows the differences between numbers as an interleaved (alternating array) of numbers. One is (4,6,8,10,...), the other at (2,4,6,8,10,...). Thus by knowing the first two numbers in the sequence, (viz. 1 and 5) the sequence can be generated. The three consecutive numbers (97, 113, 127), as their squares, are one of the diagonals of the Bremmer's square.
4. The Bremner square. however, is not magic since the sum of the right diagonal is 38307 while all other sums are 21609.


5. If we square all the numbers in the first sequence we obtain the Sequence table II. The entries (in color) between squares is the difference (Δ) between numbers in the right diagonal tuple (a², b², c²) of a magic square.

Sequence of Numbers I

1		5		7		13		17		25		31		41		49		61		71
	4		2		6		4		8		6		10		8		12		10		14
85		97		113		127		145		161		181		199		221		241		265
	12		16		14		18		16		20		18		22		20		24		22
287		313		337		365		391		421		449		481		511		545		577
	26		24		28		26		30		28		32		30		34		32		36

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Sequence of Squares II

1		25		49		169		289		625		9611
	24		24		120		120		336		336		720
1681		2401		3721		5041		7225		9409		12769
	720		1320		1320		2184		2184		3360		3360
16129		21025		25921		32761		39601		48841		58081
	4896		4896		6840		6840		9240		9240		12144
70225		82369		97969		113569		133225		152881		177241
	12144		15600		15600		19656		19656		24360		24360
			201601		231361		261121		297025		332929
				29760		29760		35904		35904

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6. If we modify Bremner's square, magic square A can be generated. Another example of a magic square (B) produced from the tuple (337, 365, 391), as their squares, also obtained from the sequence table I is shown below. The magic sum in this case is 399675.

Magic square A 582 18814 1272 25534 1132 22 972 822 22174

Magic square B 2552 181769 3912 221081 3652 2132 3372 2912 207425

7. If we take the difference of squares from above and divide each by 24 and throw out duplicates, the following sequence is found (first array). The second sequence is the difference of differences and shows that the natural number sequence of squares is generated (in yellow). So what started out as a method for generating consecutive squares led us back to the natural numbers.

The Sequence Extended

Below is shown a copy of a program used to generate the sequence with 75 and 100 entries.

function sequence(r) 
/* r is used as an argument when the function is called */
{
var a=0; var c=0;

print a 1 to start the sequence

for (n=3; n<=r;n++)  /*Cycle thru N, n starts at 3*/
{
    
   a= (n*n + 1)/2; 
     
   c= (n*n +2*n - 1)/2;
    
   print a and c

   n=n+1;


}  
}

To go back up.

This concludes the new sequence of squares. The next page treats the same topic but the entries for the right diagonals are tabulated and each row calculated using algebraic equations.
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