Recently a new method for the generation of squares of squares were produced in Part IA through Part IVE.
In addition, a new interleaved sequence of
This site will show two other sequences are generated from the square sequence method developed in Part IA and ending with Part IVE .
The previous parts showed that a tuple (a_{1},b_{1},c_{1}) can be converted into a different tuple (a,b,c) basically a transformation of the type (a_{1},b_{1},c_{1}) ⇒ (a,b,c). In addition, the initial tuples start out with the tuples (1,b_{1},1) or (1,1,c_{1}) in which the b_{1}s and the c_{1}s have the following values:
where k is any natural integer from 1 to ∞ used in calculating f and the denominator d = 2(2b_{1} − c_{1} − 1 ) in the equation:
This equation is critical is that it is the initial starting point for generating the interleaved sequences.
The tables listed below although produced according to the methods of Parts IA through IVE, were actually computed and outputted by a computer program. Two sequences are apparent as highlighted in color. The one in green, and the one in white. For example each green tuple ends with a number which is repeated in the next green tuple. Likewise for the white tuples. Initially the tables was constructed using the initial tuple (1,1,1).
Table II was inverted (c_{1} placed first and a_{1} placed last) in order that the sequence increases in the positive direction, otherwise the numbers being negative will increase to the left. Table II, however, is in the right order in Part 4E.





To separate out the three sequences and generate the equations for each, the following method was found. All we need are two equations to generate
the interleaved sequences in this case we wil take a and b.
Since two sequences may be produced from table II above, a factor f_{s} = 2 will be used to convert
Using these equations (or copying directly from Table II) we obtain for the first sequence:
1, 1, 1, 5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839
We can also convert this series to a positive one by replacing c and b by
−c and −b to obtain:
So that the sequence increases in the positive direction:
1, 1, 1, 5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839
To obtain the equations for the second sequence we use the second line of table II (1,1,5) and solve for f.
Substituting the value of 2n for n in the equations for a and b as was done above, affords the equations for the second sequence:
Using these equations (or copying directly from Table II) we obtain for the second sequence:
2, 2, 2, 10, 14, 26, 34, 50, 62, 82, 98, 122, 142, 170, 194, 226, 254, 290, 322, 362, 398, 442, 482, 530, 574, 626, 674, 730, 782, 842, 898
And again we can also convert this series to a positive one by replacing c and b by
−c and −b to obtain:
So that the sequence increases in the positive direction:
2, 2, 2, 10, 14, 26, 34, 50, 62, 82, 98, 122, 142, 170, 194, 226, 254, 290, 322, 362, 398, 442, 482, 530, 574, 626, 674, 730, 782, 842, 898
So in effect what we have done is to produce two new interleaved sequences via a completely new route. Any other way would probably have generated one sequence without the knowledge that others exist. In addition, both sequences are related via the general sequence formula (before factoring in f_{s}) and thus constitute a family of interleaved sequences.
This concludes Part F. Go back to homepage.
Copyright © 2012 by Eddie N Gutierrez. EMail: Fiboguti89@Yahoo.com