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

%I A164281 #31 Feb 18 2025 08:57:45
%S A164281 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,
%T A164281 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,
%U A164281 4,2,4,8,4,2,4,8,16,8,4,8,4,2,4,8,16,8
%N A164281 Triangle read by rows, a Petoukhov sequence (cf. A164279) generated from (1,2).
%C A164281 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.
%C A164281 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).
%C A164281 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.
%C A164281 Rows tend to A166242. - _Gary W. Adamson_, Oct 10 2009
%D A164281 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.
%H A164281 Jon Maiga, <a href="/A164281/b164281.txt">Table of n, a(n) for n = 0..1022</a> (Rows 0..9)
%F A164281 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.
%F A164281 A(n, k) = 2^(A088696(n+1, k)-1). - _Andrey Zabolotskiy_, Feb 18 2025
%e A164281 First few rows of the triangle =
%e A164281   1;
%e A164281   1, 2;
%e A164281   1, 2, 4, 2;
%e A164281   1, 2, 4, 2, 4, 8, 4, 2;
%e A164281   1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2;
%e A164281   ...
%e A164281 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).
%t A164281 A088696[n_]:=A088696[n]=Flatten[NestList[Join[#,Reverse[#]+1]&,{1},15]][[n]];
%t A164281 A164281[0]=1;
%t A164281 A164281[n_]:=If[IntegerQ[Log2[n+1]], 1, If[A088696[n+1]>A088696[n], 2*A164281[n-1], A164281[n-1]/2]]
%t A164281 Array[A164281,100,0] (* _Jon Maiga_, Oct 04 2019 *)
%Y A164281 Cf. A088696, A164279, A164056.
%Y A164281 Cf. A166242.
%K A164281 nonn,tabf
%O A164281 0,3
%A A164281 _Gary W. Adamson_, Aug 12 2009
%E A164281 Corrected and more terms from _Jon Maiga_, Oct 04 2019