A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

GEOMETRIC PROGRESSION AND RECURSION METHODS TO GENERATE NEW SEQUENCES (Part VF)

Picture of a square

Novel Geometric Progressions using the Magic Ratio (R)

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 = (1 + √2)2 = 5.8284268.... as part of the common ratio.

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.

Section I - Geometric Progression for Line 1 values

Geometric Progression for b Values

The geometric progression using the initial b = 2 is given by the sequence:

2, 2(R + 1/R), [2(R + 1/R)]*R, [2R(R + 1/R)]*R, [2R2(R + 1/R)]*R, [2R3(R + 1/R)]*R, [2R4(R + 1/R)]*R,...

where the previous value [2Rn(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:

2, 12, 70, 408, 2378, 13860, 80782, 470832, 2744210, 15994428,...

after rounding the lower values to equal the b values of line one from tables V thru XXIV.

Geometric Progression for c values

The geometric progression using the initial c = 3 is given by the sequence:

3, 3(R - 1/R), [3(R - 1/R)]*R, [3R(R - 1/R)]*R, [3R2(R - 1/R)]*R, [3R3(R - 1/R)]*R, [3R4(R - 1/R)]*R,...

where the previous value [3Rn(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:

3, 17, 99, 577, 3363, 19601, 114243, 665857, 3880899, 22619537,...

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.

Section II - Recursive Formulas for Table Lines Greater than 1

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.

Sequence of a(n) Values

Sequence Ia

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 R = 5.828427125... and the multiplier is R + 1/Rm:

a(0) = 97
a(1) = 433
a(2) = [a(1) - a(0)] * (R + 1/R2) + a(1)
a(3) = [a(2) - a(1)] * (R + 1/R4) + a(2)
a(4) = [a(3) - a(2)] * (R + 1/R6) + a(3)
a(5) = [a(4) - a(3)] * (R + 1/R8) + a(4)
.
.
a(n+1) = [a(n) - a(n-1)] * (R + 1/R2n) + a(n)

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 R2n gets bigger and bigger 1/R2n approaches zero and for all intents and purposes has little effect on R.

a(2) = 336 * 5.85786... + 433 = 2401
a(3) = 1968 * 5.82929... + 2401 = 13873
a(4) = 11472 * 5.82845... + 13873 = 80737
a(5) = 66864 * 5.828427... + 80737 = 470449
.
.

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:

δ 336 1968 11472  66864  389712 2271408 
a(n)97 433 2401 13873  80737 470449 2741857
⇒
13238736 77161008 449727312 
 15980593 93141601 542868913

Sequence IIa

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 R = 5.828427125... and the multiplier is R + 1/Rm:

a(0) = 47
a(1) = 383
a(2) = [a(1) - a(0)] * (R + 1/R2) + a(1)
a(3) = [a(2) - a(1)] * (R + 1/R4) + a(2)
a(4) = [a(3) - a(2)] * (R + 1/R6) + a(3)
a(5) = [a(4) - a(3)] * (R + 1/R8) + a(4)
.
.
a(n+1) = [a(n) - a(n-1)] * (R + 1/R2n) + a(n)

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 R2n gets bigger and bigger 1/R2n approaches zero and for all intents and purposes has little effect on R.

a(2) = 336 * 5.85786... + 383 = 2351
a(3) = 1968 * 5.82929... + 2351 = 13823
a(4) = 11472 * 5.82845... + 13823 = 80687
a(5) = 66864 * 5.828427... + 80687 = 470399
.
.

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:

δ 336 1968 11472  66864  389712 2271408 
a(n)47 383 2351 13823  80687 470399 2741807
⇒
13238736 77161008 449727312 
 15980543 93141551 542868863

Sequence of b(n) Values

Sequence Ib

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 R = 5.828427125... and the multiplier is R - 1/Rm:

b(0) = 14
b(1) = 84
b(2) = [b(1) - b(0)] * (R - 1/R2) + b(1)
b(3) = [b(2) - b(1)] * (R - 1/R4) + b(2)
b(4) = [b(3) - b(2)] * (R - 1/R6) + b(3)
b(5) = [b(4) - b(3)] * (R - 1/R8) + b(4)
.
.
b(n+1) = [b(n) - b(n-1)] * (R - 1/R2n) + b(n)

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 R2n gets bigger and bigger 1/R2n approaches zero and for all intents and purposes has little effect on R.

b(2) = 70 * 5.79898... + 84 = 490
b(3) = 406 * 5.82756... + 2401 = 2856
b(4) = 2366 * 5.82840... + 2856 = 16646
b(5) = 13790 * 5.828426... + 16646 = 97020
.
.

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:

δ 70 406 2336  13790 80374 468454 
b(n)14 84 490 2856  16646 97020 565474
⇒
2730350 15913646 92751526 
 3295824 19209470 111960996

Sequence of c(n) Values

Sequence Ic

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 R = 5.828427125... and the multiplier is R + 1/Rm:

c(0) = 99
c(1) = 449
c(2) = [c(1) - c(0)] * (R + 1/R2) + c(1)
c(3) = [c(2) - c(1)] * (R + 1/R4) + c(2)
c(4) = [c(3) - c(2)] * (R + 1/R6) + c(3)
c(5) = [c(4) - c(3)] * (R + 1/R8) + c(4)
.
.
c(n+1) = [c(n) - c(n-1)] * (R + 1/R2n) + c(n)

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 R2n gets bigger and bigger 1/R2n approaches zero and for all intents and purposes has little effect on R.

c(2) = 350 * 5.85786... + 449 = 2499
c(3) = 2050 * 5.82929... + 2499 = 14449
c(4) = 11950 * 5.82845... + 14449 = 84099
c(5) = 69650 * 5.828427... + 84099 = 490049
.
.

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:

δ 350 2050 11950  69650  405950 2366050 
c(n)99 449 2499 14449  84099 490049 2856099
⇒
13790350 80376050 468465950 
 16646449 97022499 565488449

Sequence IIc

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 R = 5.828427125... and the multiplier is R + 1/Rm:

c(0) = 51
c(1) = 401
c(2) = [c(1) - c(0)] * (R + 1/R2) + c(1)
c(3) = [c(2) - c(1)] * (R + 1/R4) + c(2)
c(4) = [c(3) - c(2)] * (R + 1/R6) + c(3)
c(5) = [c(4) - c(3)] * (R + 1/R8) + c(4)
.
.
c(n+1) = [c(n) - c(n-1)] * (R + 1/R2n) + c(n)

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 R2n gets bigger and bigger 1/R2n approaches zero and for all intents and purposes has little effect on R.

c(2) = 350 * 5.85786... + 401 = 2451
c(3) = 2050 * 5.82929... + 2451 = 14401
c(4) = 11950 * 5.82845... + 14401 = 84051
c(5) = 69650 * 5.828427... + 84051 = 490001
.
.

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:

δ 350 2050 11950  69650  405950 2366050 
c(n)51 401 2451 14401  84051 490001 2856051
⇒
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. E-Mail: enaguti1949@gmail.com