A164092 -id:A164092 - cp's OEIS frontend

A164279 Triangle of 2^n terms per row, a Petoukhov sequence generated from (3,2).

1, 3, 2, 9, 6, 4, 6, 27, 18, 12, 18, 12, 8, 12, 18, 81, 54, 36, 54, 36, 24, 36, 54, 36, 24, 16, 24, 36, 24, 36, 54

Offset: 0

Gary W. Adamson, Aug 11 2009

Row sums = powers of 5: (1, 5, 25, 125,...).
Petoukhov has pioneered the investigation of a class of matrices that are squares of other matrices composed of entirely irrational terms. A164279 terms = top rows, left columns of the Petoukhov matrices shown in A164092.
The Petoukhov matrices associated with A164279 are shown in A164092 along with their derivation from phi, 1.618033989...
The original Petoukhov matrices were in a binary Karnaugh map format.
I have standardized the matrices and sequences, mapping them on the Gray code format shown in A147995. This allows for a ("1 operation" change from one term to the next. For example, in A164279, the next term is either (3/2)*(current term) or (2/3)*(current term) depending on the corresponding positional code of A164057: (a 1 or 0).
Note the binomial frequence of terms per row: (e.g. one 27, three 18's, three 12's, and one 8) in row 3.

			The distinct terms per row are (Cf. A036561): (1; 2,3; 4,6,9; 8,12,18,27; 16,24,36,54,81;) while the codes of A164057 begin:
.
1;
1, 0;
1, 0, 0, 1;
1, 0, 0, 1, 0, 0, 1, 1;
1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1;
...
Given (1, 3, 9, 27,...) as leftmost row terms and following the operational rules: (multiply current term by (3/2) if the corresponding code = 1; (or by (2/3) if 0). This generates A164279: .
1;
3, 2;
9, 6, 4, 6;
27, 18, 12, 18, 12, 8, 12, 18;
81, 54, 36, 54, 36, 24, 36, 54, 36, 24, 16, 24, 36, 24, 36, 54;
...

Sergei Petoukhov & Matthew He, "Symmetrical Analysis Techniques for Genetic Systems and Bioinformatics - Advanced Patterns and Applications"; IGI Global, 978-1-60566-127-9, October, 2009; Chapters 2, 4, and 6.

Cf. A036561, A164057, A147995.

Using the row terms of A036562 (a 2x3 multiplication table): (1, 3,2; 4,6,9;, 8,12,18,27;...), rows of A164279 have leftmost terms extracting the power of 9 from A036562: (1, 3, 9, 27,...). Then accessing the corresponding row codes from A164057, and starting from the left, first term = a power of 9, then given the codes of A164057 (0 or 1), the next row term of A164279 = (3/2)*current term) if the corresponding term of A164057 = 1, and (2/3)*current term if 0.

A164516 Infinite set of Petoukhov 2^n x 2^n Petoukhov matrices by antidiagonals, generated from w = (-.5 + sqrt(-3)/2).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 14 2009

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.

			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.

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.

Cf. A164092, A119633.

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.

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A164279 Triangle of 2^n terms per row, a Petoukhov sequence generated from (3,2).

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