NEW FAMILY OF SEQUENCES

THE GENERATION OF NEW SEQUENCES (Part A)

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 and are part of the sequence of numbers included in Part A (this page), Part B, Part C and Part D.

This site will show that these four sequences as well as an infinitude of others 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:

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

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.

Separation of Sequences

To separate out the two 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 a ⇒ new a and b ⇒ new b. This is accomplished by substituting the value of 2n for n in the equations for a and b, to give the equations for the first sequence:

Using these equations (or copying directly from Table II) we obtain for the first sequence whose Sloane number is listed as A216844 is:

2, 2, -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, 962, 1022, 1090, 1154, 1226, 1294, 1370, 1442, 1522, 1598, 1682, 1762, 1850, 1934, 2026

When each entry is divided by 2 the sequence is identical to the Sloane number 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, 647, 685, 721, 761, 799, 841, 881, 925, 967, 1013

To obtain the equations for the second sequence having the Sloane number A214345, we use the second line of table II (1,3,1) 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:

-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, 901, 969

So in effect what we have done is to produce two new sequences via a completely new route (the Sloane number A214345 is already known). Previous methods would have generated one sequence without the knowledge that others existed. 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 A. Go back to homepage.