The tables of partial imaginary tuples listed in Part IB, Part IC, Part IIIC, Part ID and Part IE are useful, when squared, as right diagonals for magic square of squares. I will show that new recursive formulas can be used to generate new integer sequences from the tuple numbers a, b and c using without employing R as part of the multiplier as was done previously in Part VF.
This section will use the values of a. b and c from line 8 of tables V thru XXIV. It has been found that recursive formulas for the sequences produce linear sequences without resorting to the use of R as part of the multiplier as was shown in Part VF. The indeces x(0) and x(1) are initially given and from these two values the sequences are built up in a recursive fashion. Note that x is a placeholder for a, b or c. The differences (δs) between values of both sets of a and c sequences and the average of the as and cs shown below are identical. Only the a(n) and c(n) values are similar but not quite identical. In addition, since the b values are identical in both tables only one sequence is obtained.
The recursive formulas uses a factor of 6 to multiply to the first number in each formula line. It appears that this factor is obtained from the ratio b(1)/b(0) since the same number pops up in each of the other lines in the tables V/VII or VI/VIII. A second constant k_{a} or k_{c} is added or subtracted to the recursive formulas of the as or the cs sequences but none is added/subtracted to the b sequences. In addition, if the values of either the a or c sequences are averaged, no constant is required and the formulas are structually similar to those of b (see Average Sequence Ia + IIa or Average Sequence Ic + IIc).
The reason the a or c are separated out into separate tables is that they are part of well formed tuples giving the requisite correct square values for the diagonals. If we average out the a or c values, on the other hand, non well formed tuples are generated, i.e., at least one number is a non integer square and, thus, is not suitable for use as diagonals in squares of magic squares. In other words, the equation:
is not satisfied when the averages of a and/or c are used.
From tables V and VII, line 8 the a^{} values of 97 and 433 are initialized followed by generation of a new sequence using the following recursive equations:
where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1 and k_{a} = [(aI(n) − aII(n)]*2. The aI(n) − aII(n) being the difference between two numbers having the same offset taken from each of the sequences of Table V and Table VI, viz., aI(0) − aII(0) = 97 − 47 = 50. In addition, every aI(n) − aII(n) has the same value.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (a(n)) in the second row with the sequence continuing after the arrow:

⇒ 
13238736  77161008  449727312  
15980593  93141601  542868913 
From tables VI and VIII, line 8 the a^{} values of 47 and 383 are initialized followed by generation of a new sequence :
where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1 and k_{a} = [(aI(n) − aII(n)]*2. The aI(n) − aII(n) being the difference between two numbers having the same offset taken from each of the sequences of Table V and Table VI, viz., aI(0) − aII(0) = 97 − 47 = 50. In addition, every aI(n) − aII(n) has the same value.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (a(n)) in the second row with the sequence continuing after the arrow:

⇒ 
13238736  77161008  449727312  
15980543  93141551  542868863 
A new sequence is also obtained from the averages of sequence Ia and IIa where the initialization values are 72 and 408 by followed by generation of a new sequence:
where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (a(n)) in the second row with the sequence continuing after the arrow:

⇒ 
13238736  77161008  449727312  
15980568  93141576  542868888 
From tables V and VII, line 8 the b^{} values of 14 and 84 are initialized followed by generation of a new sequence using the following recursive formulas:
where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (b(n)) in the second row with the sequence continuing after the arrow:

⇒ 
2730350  15913646  92751526  
3295824  19209470  111960996 
From tables V and VII, line 8 the c^{} values of 99 and 449 are initialized followed by generation of a new sequence using the following recursive formulas:
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1 and k_{c} = [(cI(n) − cII(n)]*2. The cI(n) − cII(n) being the difference between two numbers having the same offset taken from each of the sequences of Table V and Table VI, viz., cI(0) − cII(0) = 99 − 51 = 48. In addition, every cI(n) − cII(n) has the same value.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (c(n)) in the second row with the sequence continuing after the arrow:

⇒ 
13790350  80376050  468465950  
16646449  97022499  565488449 
From table VI and VIII, line 8 the c^{} values of 51 and 401 are initialized followed by generation of a new sequence using the following recursive formulas:
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1 and k_{c} = [(cI(n) − cII(n)]*2. The cI(n) − cII(n) being the difference between two numbers having the same offset taken from each of the sequences of Table V and Table VI, viz., cI(0) − cII(0) = 99 − 51 = 48. In addition, every cI(n) − cII(n) has the same value.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (c(n)) in the second row with the sequence continuing after the arrow:

⇒ 
13790350  80376050  468465950  
16646401  97022451  565488401 
A new sequence is also obtained from the averages of sequence Ic and IIc where the initialization values are 75 and 425 by followed by generation of a new sequence:
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1.
The sequence up to 10 values is shown as follows and can be compared to the originals from tables V thru XXIV. Note that the first row corresponds to the differences (δs) between the numbers (c(n)) in the second row with the sequence continuing after the arrow:

⇒ 
13790350  80376050  468465950  
16646425  97022475  565488425 
This concludes Part VIF using two recursive progressions to generate sequences identical to those generated in tables V thru XXIV. Finally to see what started it all go to Part IG which lists the new set of magic square of seven squares.
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Copyright © 2016 by Eddie N Gutierrez. EMail: enaguti1949@gmail.com