NEW FAMILY OF SEQUENCES

THE GENERATION OF NEW SEQUENCES (Part G)

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:

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. 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).

Only One Sequence

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

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 a_n 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.