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 in Section I that b and c of line one, Table V may be used to generate,
using two novel geometric progressions, all the values found in tables VI thru XXIV
via successive multiplications using R
=
I will also generate in Section II the sequences of lines greater than one for the tuple numbers a, b and c using new recursive formulas employing R as part of the multiplier.
The geometric progression using the initial b = 2 is given by the sequence:
where the previous value [2R^{n}(R + 1/R)] is multiplied by R to get the current value. Notice that when n is 0 the value corresponds to the second value in the sequence. Inserting the value for R affords the following sequence:
after rounding the lower values to equal the b values of line one from tables V thru XXIV.
The geometric progression using the initial c = 3 is given by the sequence:
where the previous value [3R^{n}(R  1/R)] is multiplied by R to get the current value. Notice that when n is 0 the value corresponds to the second value in the sequence. Inserting the value for R affords the following sequence:
after rounding the lower values to equal the c values of line one from tables V thru XXIV. The formula for c, however, only works for the c of line one and for none of the other a and c greater than one, unlike the b values which are still operable.
This section will use the values of a. b and c from line 8 of tables V thru XXIV. The formula for the sequences of lines greater than 1 are recursive sequences employing R as part of the multiplier. 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 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.
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
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1. It is also to be noted that as R^{2n} gets bigger and bigger 1/R^{2n} approaches zero and for all intents and purposes has little effect on R.
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
using the following recursive formulas where
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1. It is also to be noted that as R^{2n} gets bigger and bigger 1/R^{2n} approaches zero and for all intents and purposes has little effect on R.
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 
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
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1. It is also to be noted that as R^{2n} gets bigger and bigger 1/R^{2n} approaches zero and for all intents and purposes has little effect on R.
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 where
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1. It is also to be noted that as R^{2n} gets bigger and bigger 1/R^{2n} approaches zero and for all intents and purposes has little effect on R.
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 where
and where the general formula for the sequences is the last line of the recursive formulas with n ≥ 1. It is also to be noted that as R^{2n} gets bigger and bigger 1/R^{2n} approaches zero and for all intents and purposes has little effect on R.
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 
This concludes Part VF using the two geometric progressions as well as 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.
To go to recursive seuqnces without the use of R see Part VIF. Go back to homepage.
Copyright © 2016 by Eddie N Gutierrez. EMail: enaguti1949@gmail.com