NEW FAMILY OF SEQUENCES

THE GENERATION OF NEW SEQUENCES (Part F)

Picture of a square

Introduction

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 numbers was developed from this work and the sequence awarded the Sloane number A178218. OEIS has also published four other numbers based on this particular type of sequence. These numbers are A214345, A214493, A214393 and A214405.

This site will show two other sequences are generated from the square sequence method developed in Part IA and ending with Part IVE .

Some Background

The previous parts showed that a tuple (a1,b1,c1) can be converted into a different tuple (a,b,c) basically a transformation of the type (a1,b1,c1) ⇒ (a,b,c). In addition, the initial tuples start out with the tuples (1,b1,1) or (1,1,c1) in which the b1s and the c1s have the following values:

b1 =  k
b1 = −k
c1 =  k
c1 = −k

where k is any natural integer from 1 to ∞ used in calculating f and the denominator d = 2(2b1 − c1 − 1 ) in the equation:

f = [2e2n2 + (4c1 − 4b1) en +(1 − 2b12 + c12)] / {2(2b1 − c1 − 1)}

This equation is critical is that it is the initial starting point for generating the interleaved sequences.

Table and General Sequence

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 (c1 placed first and a1 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.

  
n
0
1
2
3
4
5
6
7
8
9
10
11
12
Table I
a1 b1 c1
1-11
115
139
1513
1717
1921
11125
11329
11533
11737
11941
12145
12349
  
f = S/d
0
-3
-8
-15
-24
-35
-48
-63
-80
-99
-120
-143
-168
Table II
c b a
1-11
2-2-2
1-5-7
-2-10-14
-7-17-23
-14-26-34
-23-37-47
-34-50-62
-47-65-79
-62-82-98
-79-101-119
-98-122-142
-119-145-167
  
Δ
0
0
-24
-96
-240
-480
-840
-1344
-2016
-2880
-3960
-5280
-6864
For the tuple (1,−1,1)   f is calculated to be:
f = [2e2n2 + (4 + 4)en + 0]/2×(−4) = [2e2n2 − 8en]/−8
Setting e = 2 and g = 4 affords f = −n2 −2n
Substituting this value of f in
(f + 1, f + 2n − 1, f + 4n + 1) as was shown in Part IA
gives the general equations for the complete sequence (after inversion of a and c)

c = (−n2 − 2n + 4n + 1 ) = (−n2 + 2n + 1 )
b = (−n2 − 2n + 2n − 1 ) = (−n2 − 1)
a = (−n2 − 2n + 1 ) = (−n2 − 4n + 1)

Separation of Sequences

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 fs = 2 will be used to convert c ⇒ new c and b ⇒ new b. This is accomplished by substituting the value of 2n for n in the equations for c and b, to give the equations for the first sequence:

c = (−4n2 + 4n + 1 )
b = (−4n2 − 1)

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:

c = (4n2 − 4n − 1 )
b = (4n2 + 1)

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.

For the tuple (1,1,5)   f is calculated to be:
f = [2e2n2 + (20 − 4)en + 24]/2×−4 = [2e2n2 16en − 24]/−8
and by setting e = 2 affords the general equation f = −n2 − 4n − 3
Substituting this value of f in
(f + 1, f + 2n + 1, f + 4n + 1) as was shown in Part IA
gives the general equations for the complete sequence starting at point (1,1,5) (after inversion of a and c)

c = (−n2 − 4n − 3 + 4n + 5 ) = (−n2 + 2 )
b = (−n2 − 4n − 3 + 2n + 1 ) = (−n2 − 2n − 2)
a = (−n2 − 4n − 3 + 1) = (−n2 − 4n − 2)

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:

c = (−4n2 + 2 )
b = (−4n2 − 4n − 2)

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:

c = (4n2 − 2 )
b = (4n2 + 4n + 2)

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 fs) and thus constitute a family of interleaved sequences.

This concludes Part F. Go back to homepage.

Copyright © 2012 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com