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, 243, 162, 108, 162, 108, 72, 108, 162, 108, 72, 48, 72, 108, 72, 108, 162, 108, 72, 48, 72, 48, 32, 48, 72, 108, 72, 48, 72, 108, 72, 108, 162

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 term 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

Zhuorui He, Table of n, a(n) for n = 0..8190 (first 13 rows, flattened)

Formula

Using the row terms of A036561 (a 2x3 multiplication table): (1, 2,3; 4,6,9; 8,12,18,27;...), rows of A164279 have leftmost terms extracting the power of 3 from A036561: (1, 3, 9, 27,...). Then accessing the corresponding row terms from A164057, and starting from the left, first term = a power of 3, then given the terms 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. [Corrected by Zhuorui He, Mar 04 2026]

From Zhuorui He, Mar 04 2026: (Start)

T(0, 0) = 1, T(n, m) = 3*T(n-1, m) if m < 2^(n-1), else T(n, m) = 2*T(n-1, 2^n-m-1).

T(n, m) = 3^n * (2/3)^(A088696(n, m)-1).

As a sequence: a(n) = 3^A000523(n+1)*(2/3)^(A088696(n+1)-1). (End)

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 terms 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, 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 term = 1; (or by (2/3) if 0).
This generates this sequence:
  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

Cf. A036561, A164057, A147995, A000523, A088696.

Sequence in context: A350831 A140590 A329211 * A289053 A191539 A235539

Adjacent sequences: A164276 A164277 A164278 * A164280 A164281 A164282

Keyword

nonn,tabf

Author

Gary W. Adamson, Aug 11 2009

Extensions

More terms from Zhuorui He, Mar 04 2026

Status

approved