A197819 Table of binary Walsh functions w(A001317), columns read as binary numbers.

0, 1, 0, 3, 2, 1, 0, 15, 10, 5, 12, 3, 6, 9, 8, 7, 2, 13, 4, 11, 14, 1, 0, 255, 170, 85, 204, 51, 102, 153, 136, 119, 34, 221, 68, 187, 238, 17, 240, 15, 90, 165, 60, 195, 150, 105, 120, 135, 210, 45, 180, 75, 30, 225, 160, 95, 10, 245, 108, 147

Offset: 0

Author

Tilman Piesk, Oct 18 2011, reviewed Aug 25 2013

Keywords

• nonn
• tabf

Comments

The rows of an infinite binary Walsh matrix (compare A228539) are the binary Walsh functions w(0),w(1),w(2),w(3),...

This number triangle represents the infinite binary array w(1),w(3),w(5),w(15),... (1,3,5,15,... is A001317.)

T(n,k) is column k of the (2^n) X (2^2^n) submatrix read as a binary number.

Top left 4 X 16 submatrix of the binary array:

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0

0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

In the sequence this is represented by row 2:

0 15 10 5 12 3 6 9 8 7 2 13 4 11 14 1

A195467 is the infinite array of Gray code permutation powers. It can be defined by this binary array, which happens to be A195467 mod 2.

Each odd column is the complement of the even column on its left.

Each row of the number triangle is a self-inverse Walsh permutation. The subsequence of even entries (on the even places) divided by 2 is a self-inverse Walsh permutation too.

Links

• Tilman Piesk, Rows 0..3 of the triangle, flattened
• Tilman Piesk, permutation matrices of row 2 and row 3
• Tilman Piesk, Explanations (including the 8x256 submatrix) and MATLAB code showing the connection with A195467

Crossrefs

Cf. A195467 (consecutive powers of the Gray code permutation).

Cf. A001317 (Sierpinski triangle rows read like binary numbers).