A164281 Triangle read by rows, a Petoukhov sequence (cf. A164279) generated from (1,2).

1, 1, 2, 1, 2, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8, 16, 32, 16, 8, 4, 8, 16, 8, 4, 8, 4, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2, 4, 8, 16, 8

Offset

0,3

Comments

Row sums = powers of 3: (1, 3, 9, 27, 81, ...). A164279 = a Petoukhov sequence generated through analogous principles from (3,2), with row sums = powers of 5.

Essentially, A164281 converts the terms (1,2,4,8,...) into rows with a binomial distribution as to frequency of terms. For example, row 3 has one 1, three 2's, three 4's, and one 8. This property arises due to the origin of the system of codes in A164056 (derived from the Gray code).

A Gray code origin also preserves the "one bit" (in this case, a "one product operation") since in each row, the next term is either twice current term or (1/2) current term.

Rows tend to A166242. [Gary W. Adamson, Oct 10 2009]

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

Jon Maiga, Table of n, a(n) for n = 0..1022 (Rows 0..9)

Formula

Given row terms of triangle A059268: (1; 1,2; 1,2,4; 1,2,4,8;...) and the digital codes in A164056: (0; 0,1; 0,1,1,0; 0,1,1,0,1,1,0,0;...); beginning with "1" in each row, multiply by 2 to obtain the next term to the right, if the corresponding positional term in A164056 = "1". Divide by 2 if the corresponding A164056 term = 0.

Example

First few rows of the triangle =
1;
1, 2;
1, 2, 4, 2;
1, 2, 4, 2, 4, 8, 4, 2;
1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2;
...
Example: row 3 of A164056 =
(0, 1, 1, 0, 1, 1, 0, 0), so beginning with "1" at left, row 3 of A164281 = (1, 2, 4, 2, 4, 8, 4, 2).

Mathematica

A088696[n_]:=A088696[n]=Flatten[NestList[Join[#, Reverse[#]+1]&, {1}, 15]][[n]];
A164281[0]=1;
A164281[n_]:=If[IntegerQ[Log2[n+1]], 1, If[A088696[n+1]>A088696[n], 2*A164281[n-1], A164281[n-1]/2]]
Array[A164281, 100, 0] (* Jon Maiga, Oct 04 2019 *)

Crossrefs

Cf. A088696, A164279, A164056.

Cf. A166242 [Gary W. Adamson, Oct 10 2009]

Sequence in context: A273917 A186187 A013943 * A082693 A225081 A307368

Adjacent sequences: A164278 A164279 A164280 * A164282 A164283 A164284

Keyword

nonn,tabf

Author

Gary W. Adamson, Aug 12 2009

Extensions

Corrected and more terms from Jon Maiga, Oct 04 2019

Status

approved