A350289 Infinite binary Walsh matrix read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0

Offset

0

Comments

The binary Walsh matrix of order 2^n, using natural ordering, is the 2^n-th principal submatrix of this matrix.

This sequence begins to diverge from A219463 at n=24, corresponding to (i,j)=(3,3).

Links

Table of n, a(n) for n=0..104.

Eric Weisstein's World of Mathematics, Walsh Function

Wikipedia, Walsh matrix

Formula

A(i,j) = A010060(A004198(i,j)) = hammingweight(i AND j) mod 2.

Example

Top left corner of infinite binary Walsh matrix:
  0 0 0 0 0 0 0 0
  0 1 0 1 0 1 0 1
  0 0 1 1 0 0 1 1
  0 1 1 0 0 1 1 0
  0 0 0 0 1 1 1 1
  0 1 0 1 1 0 1 0
  0 0 1 1 1 1 0 0
  0 1 1 0 1 0 0 1

Mathematica

Flatten[Table[
  Mod[DigitCount[BitAnd[k, n - k], 2, 1], 2], {n, 0, 14}, {k, 0, n}]]

Prog

(PARI) A(i, j) = hammingweight(bitand(i, j)) % 2

Crossrefs

Cf. A197818 (negated antidiagonals as decimal), A228539, A228540.

Sequence in context: A354031 A354035 A025457 * A219463 A286688 A356923

Adjacent sequences: A350286 A350287 A350288 * A350290 A350291 A350292

Keyword

nonn,tabl

Author

Jeremy Tan, Dec 23 2021

Status

approved