|
| |
|
|
A164279
|
|
Triangle of 2^n terms per row, a Petoukhov sequence generated from (3,2).
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
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.
|
|
|
REFERENCES
|
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.
|
|
|
LINKS
|
|
|
|
FORMULA
|
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.
|
|
|
EXAMPLE
|
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;
...
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
