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A164575 Antidiagonals of an infinite set of 2^n x 2^n Petoukhov matrices generated from (7,2). 1
7, 2, 2, 49, 14, 14, 4, 49, 4, 14, 14, 14, 14, 4, 49, 4, 14, 14, 49, 343, 98, 98, 28, 343, 28, 98, 98, 98, 98, 28, 28, 343, 28, 28, 8, 8, 98, 98, 8, 8, 28, 28, 28, 343, 28, 28, 28, 98, 98, 98, 98, 98, 98, 98, 98, 28, 28, 28, 343, 28, 28, 28, 8, 8, 98, 98, 8, 8, 28, 28, 343, 28, 28 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The terms selected for usage in the 2^n x 2^n matrices are found in the multiplication table for 7^n x 2^n; then refer to antidiagonals:

.

...1.....2.....4.....8.....16;

...7....14....28....56.......;

..49....98...196.............;

.343...686...................;

2401.........................;

.

The 2 X 2 matrix is composed of terms (7,2); the 4 X 4 matrix uses terms (49, 14, and 4), while the 8 X 8 matrix uses terms (343, 98, 28, 8) with a binomial frequency.

We can recreate the top row and left columns of the 8 X 8 matrix using (343, 98, 28, and 8) and starting with 343. Given the code (row 3, A164309): (1, 0, 0, 1, 0, 0, 1, 1), multiply current term by (2/7) if corresponding term = 0. Multiply current term by (7/2) if next term = 1. We obtain

.

..1....0....0....1....0....0....1....1; =

343...98...28...98...28...08...28...98; the same string as obtained by other methods.

Contribution from Gary W. Adamson, Aug 23 2009: (Start)

The subset of 2 X 2 matrices with k = powers of phi, (1.6180339...) are by analogous procedures based on (3,2), (7,2), (18,2)...; where (3, 7, 18, 47,... are the bisected Lucas numbers of A005248 starting with offset 1. Let Q = [phi^n, 1/phi^n; phi^n, 1/phi^n], then Q^2 = [A005248(n), 2; 2, A005248(n)]; where the first 3 2 X 2 matrices of Q^n = [3,2; 2,3], [7,2; 2,7], and [18,2; 2,18]. (End)

REFERENCES

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

LINKS

Table of n, a(n) for n=1..73.

FORMULA

Several methods are presented analogous to the (5,2) case in A164557: all related to mapping a certain class of constants k, (roots to x^4 - Nx^2 + 1 = 0) onto Gray code maps. The 2 X 2 matrix [7,2; 2,7] has a square root given by x^4 - 7x^2 + 1, k = phi^2 (2.618033989...,) and 1/k.

Given the exponent codes of A164092:

  0;

  1, -1;

  2,  0, -2, 0;

  3,  1, -1, 1, -1, -3, -1, 1;

.

we create alternating circulant matrices with such strings as the top row and left column as shown in A164557. The 4 X 4 matrix is thus (exponents to k):

.

   2,  0, -2,  0;

   0,  2,  0, -2;

  -2,  0,  2,  0;

   0, -2,  0,  2;

.

and with k = phi^2 we get matrix P:

.

   phi^4,     1,    1/phi^4,    1;

     1,     phi^4,     1,    1/phi^4;

  1/phi^4,    1,     phi^4,     1;

     1,    1/phi^4,    1,     phi^4;

.

Then P^2 =

.

  49, 14, 04, 14;

  14, 49, 14, 04;

  04, 14, 49, 14;

  14, 04, 14, 49;

.

Using the alternate circulant method for the 8 X 8 matrix, the "A" sequence of exponents (Cf. A164092) = [3, 1, -1, 1, -1, -3, -1, 1] = top row and left column of the 8 X 8 matrix, as to exponents of k. Diagonal = all 3's. Then given upper left term of the matrix = (1,1), we circulate odd columns from position (n,n) downwards using the "A" Sequence. Circulate from (n,n) -> upwards if the column is even. This generates the 8 X 8 exponent matrix shown in A164516 = R. Then square R, getting the 8 X 8 matrix R^2 shown in the example section.

Given the multiplication codes of A164309:

  1;

  1, 0;

  1, 0, 0, 1;

  1, 0, 0, 1, 0, 0, 1, 1;

  1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1;

.

we may recreate the top rows of each matrix, for example, the 8 X 8 matrix: For the code of length 2^n, we begin with the integer 7^3 at left; then if the next corresponding code term = 0, multiply current term by (2/7). If the next code term = 1, multiply the current term by (7/2). With n = 3; we have the 8 bit string ...1....0....0....1....0....0....1....1; then following the rules, we get: 343...98...28...98...28...08...28...98; reproducing the circulant terms for the 8 X 8 matrix.

We may obtain the same set of 8 terms by mapping (7 & 2) on the top row of the DNA codon map shown in A147995, and using the conversion rules (C,G)=7; (A,U)=2. Or, using bits from top -> down, (0,0; 1,1)=3; (0,1; 1,0)=2, then multiply the terms. We obtain:

.

000...001...011...010...110...111...101...100;

000...000...000...000...000...000...000...000;

777...772...722...727...227...222...272...277; =

343...098...028...098...028...008...028...098; = top row of the 8 X 8 matrix.

.

The 8 X 8 matrix can then be generated using the circulant rule: Let the 8 term string = "A", then put "A" as top row and left column. Diagonal = all 343's; then circulate odd labeled columns from position (n,n) down, while even columns are circulated from (n,n) upwards, given upper left term = (1,1).

EXAMPLE

The 8x8 matrix R^2 =

.

  343,...98...28...98...28....8...28...98;

  .98,..343...98...28...08...28...98,..28;

  .28....98..343...98...28...98...28...08;

  .98....28...98..343...98...28...08...28;

  .28....08...28...98..343...98...28...98;

  .08....28...98...28...98..343...98...28;

  .28....98...28...08...28...98..343...98;

  .98....28...08...28...98...28...98..343;

.

Sequence A164575 = antidiagonals of the 2^n X 2^n matrices, exhausting all terms in each matrix before going onto the next matrix.

CROSSREFS

Cf. A147995, A164557, A164522, A164092, A164282, A164516, A164309, A164057.

Cf. A005248.

Sequence in context: A300304 A208647 A163981 * A126341 A078087 A176976

Adjacent sequences:  A164572 A164573 A164574 * A164576 A164577 A164578

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 16 2009

STATUS

approved

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Last modified March 26 18:36 EDT 2019. Contains 321511 sequences. (Running on oeis4.)