1,1

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 2x2 matrix is composed of terms (7,2); the 4x4 matrix uses term (49, 14,

and 4); while the 8x8 matrix uses terms (343, 98, 28, 8); with a binomial

frequency.

We can recreate the top row and left columns of the 8x8 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 (qntmpkt(AT)yahoo.com), Aug 23 2009: (Start)

The subset of 2x2 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 2x2 matrices of Q^n = [3,2; 2,3], [7,2; 2,7],

and [18,2; 2,18]. (End)

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.

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 2x2 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 4x4 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 8x8 matrix, the "A" sequence of

exponents (Cf. A164092) = [3, 1, -1, 1, -1, -3, -1, 1] = top row and left

column of the 8x8 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 8x8 exponent matrix shown

in A164516 = R. Then square R, getting the 8x8 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 8x8 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 cirulant terms for

.

the 8x8 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 8x8 matrix.

.

The 8x8 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

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.

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

A005248 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 23 2009]

Sequence in context: A010141 A154759 A163981 this_sequence A126341 A078087 A021857

Adjacent sequences: A164572 A164573 A164574 this_sequence A164576 A164577 A164578

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 16 2009