NEW FAMILY OF SEQUENCES

THE GENERATION OF NEW SEQUENCES (Part G)

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. This page updates this sequence so that it follows the method developed here. 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(2b1c1 − 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. This page is a summary of the initial work done on 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. Only one sequence is apparent. One tuple ends and the other continues on the next line. Initially the tables was constructed using the initial tuple (1,1,-1).

  
n
0
1
2
3
4
5
6
7
8
9
10
11
12
Table I
a1 b1 c1
11-1
133
157
1711
1915
11119
11323
11527
11731
11935
12139
12343
12547
  
f = S/d
0
-2
0
6
16
30
48
70
96
126
160
198
240
Table II
a b c
11-1
-111
157
71317
172531
314149
496171
718597
97113127
127145161
161181199
199221241
241265287
  
Δ
0
0
24
120
336
720
1320
2184
3360
4896
6840
9240
12144
For the tuple (1,1, −1)   f is calculated to be:
f = [2e2n2 + (− 4 − 4)en + 0]/2×2 = [2e2n2 − 8en + 0]/4
Setting e = 2 and g = 4 affords f = 2n2 − 4n
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

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

Only One Sequence

No separation of sequences has to be done. Since the factor fs = 1, the equations to be used are:

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

Using these equations (or copying directly from Table II) we obtain for the sequence whose Sloane number is A178218:

1, 1, -1, 1, 1, 5 , 7, 13, 17, 25, 31, 41, 49, 61, 71, 85, 97, 113, 127, 145, 161, 181, 199, 221, 241, 265, 287, 313, 337, 365, 391, 421, 449, 481, 511, 545, 577, 613
The Generating Function (G.f.) for this sequence is (1 − x − 3x2 + 5x3) /(1+x)(1−x)3 and differs from the Sloane number A178218 because of the four extra digits at the start of the sequence. In addition, the two equations used in A178218 are b and c, while the ones used here are a and b.

Note that a G.f. as defined by Wikepedia is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers, i.e., the coefficients of the equation are each of the terms in the sequence being looked at which in this case is :
1 + x − x2 + x3 + x4+ 5x5 + 7x6 + 13x7 + 17x8 + 25x9 + 31x10 ... and are obtained by dividing the numerator by the denominator above.


This concludes Part G. Go back to homepage.

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