1,2

Sergei Petoukhov has pioneered the investigation of certain matrices whose

square roots are irrational numbers; and in recognition his discoveries

such matrices and their accompanying sequences may be termed "Petoukhov

matrices/sequences".

Refer to A119633 for a related sequence.

Sergey Petoukhov & Matthew He, "symmetrical Analysis Techniques for Genetics systems and Bioinformatics, Advanced Patterns & Applications", IGI Global, 978-1-60566-127-9, October, 2009, Chapters 2, 4, and 6.

Given w = (-.5 + sqrt(-3)/2), use the exponent codes of A164092 to create

alternating circulant matrices such that a row with 2^n terms generates

2^n x 2^n matrices. Terms in these matrices = exponents for w, then

square the matrices. Sequence A164516 = antidiagonals of the infinite set

of 2^n x 2^n matrices, exhausting terms in the n-th matrix before using

the terms of the next matrix.

The exponent codes of A164092 are:

.

0; (skip as trivial);

1, -1; (creates the 2x2 matrix [w,1/w; 1/w,w](exponents of w = 1 & -1).

2, 0, -2, 0;

3, 1, -1, 1, -1, -3, -1, 1;

4, 3, .0, 2, .0, -2, .0, 2, 0, -2, -4, -2, 0, -2, 0, 2;

...

Exponent codes (above) are generated by adding "1" to each term in n-th row

bringing down that subset as the first half of the next row. Second half

of the next (n+1)-th) row is created by reversing the terms of n-th row

and subtracting "1" from each term. (2, 0, -2, 0) becomes (3, 1, -1, 1) as

the first half of the next row. Then append (-1, -3, -1, 1), getting

(3, 1, -1, 1, -1, -3, -1, 1) as row 3. Let these rows = "A" for each matrix

.

In a 2^n * 2^n matrix with a conventional upper left term of (1,1), place

A as the top row and left column. Put leftmost term of A into every (n,n)

(i.e. diagonal position). Then, odd columns are circulated from position

(n,n) downwards while even columns circulate upwards starting from (n,n).

Using A with 8 terms we obtain the following 8x8 matrix:

.

3, 1, -1, 1, -1, -3, -1, 1;

1, 3, 1, -1, -3, -1, 1, -1;

-1, 1, 3, 1, -1, 1, -1, -3;

1, -1, 1, 3, 1, -1, -3, -1;

-1, -3, -1, 1, 3, 1, -1, 1;

-3, -1, 1, -1, 1, 3, 1, -1;

-1, 1, -1, -3, -1, 1, 3, 1;

1, -1, -3, -1, 1, -1, 1, 3;

.

The foregoing terms are exponents to w, so our new matrix becomes:

.

1, w, 1/w, w, 1/w, 1, 1/w, w;

w, 1, w, 1/w, 1, 1/w, w, 1/w;

1/w, w, 1, w, 1/w, w, 1/w, 1;

w, 1/w, w, 1, w, 1/w, 1, 1/w;

1/w, 1, 1/w, w, 1, w, 1/w, w;

1, 1/w, w, 1/w, w, 1, w, 1/w;

1/w, w, 1/w, 1, 1/w, w, 1, w;

w, 1/w, 1, 1/w, w, 1/w, w, 1;

.

Let the foregoing matrix = Q, then take Q^2 =

.

-1, 2, -4, 2, -4, 8, -4, 2;

2, -1, 2, -4, 8, -4, 2, -4;

-4, 2, -1, 2, -4, 2, -4, 8;

2, -4, 2, -1, 2, -4, 8, -4;

-4, 8, -4, 2, -1, 2, -4, 2;

8, -4, 2, -4, 2, -1, 2, -4;

-4, 2, -4, 8, -4, 2, -1, 2;

2, -4, 8, -4, 2, -4, 2, -1;

.

Following analogous procedures for the 2x2 and 4x4 matrices, those are

[ -1, 2; 2,-1], and

.

1, -2, 4, -2;

-2, 1, -2, 4;

4, -2, 1, -2;

-2, 4, -2, 1;

.

Take antidiagonals of the matrices until all terms in each matrix are used.

A164092, A119633

Sequence in context: A106180 A055091 A014678 this_sequence A016533 A122915 A030298

Adjacent sequences: A164513 A164514 A164515 this_sequence A164517 A164518 A164519

tabl,sign

Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 14 2009