A328568 Irregular triangle read by rows; for n >= 0, the n-th row corresponds to the elements of the set {(n-k) XOR k, k = 0..n}, in ascending order (where XOR denotes the bitwise XOR operator).

0, 1, 0, 2, 3, 0, 2, 4, 1, 5, 0, 4, 6, 7, 0, 4, 6, 8, 1, 5, 9, 0, 2, 4, 8, 10, 3, 11, 0, 2, 8, 10, 12, 1, 9, 13, 0, 8, 12, 14, 15, 0, 8, 12, 14, 16, 1, 9, 13, 17, 0, 2, 8, 10, 12, 16, 18, 3, 11, 19, 0, 2, 4, 8, 10, 16, 18, 20, 1, 5, 9, 17, 21, 0, 4, 6, 8, 16, 20, 22

Offset: 0

Author

Rémy Sigrist, Oct 20 2019

Keywords

• nonn
• tabf
• look
• base

Comments

For any n >= 0, the n-th row:

- has sum A328565(n),

- has apparently length A002487(n+1),

- has first element A135481(n),

- has last element n.

Examples

			Table begins:
    0;
    1;
    0, 2;
    3;
    0, 2, 4;
    1, 5;
    0, 4, 6;
    7;
    0, 4, 6, 8;
    1, 5, 9;
    0, 2, 4, 8, 10;
    3, 11;
    0, 2, 8, 10, 12;
    1, 9, 13;
    0, 8, 12, 14;
    ...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..9851

Crossrefs

Cf. A326819 (AND variant), A326820 (OR variant).

Cf. A002487, A135481, A328565.

Programs

Maple

T:= n-> sort([{seq(Bits[Xor](n-k, k), k=0..n)}[]])[]:
seq(T(n), n=0..30);  # Alois P. Heinz, Oct 20 2019

Mathematica

Union /@ Table[BitXor[n - k, k], {n, 0, 22}, {k, 0, n}] // Flatten (* George Beck, Jun 09 2023 *)

PARI

row(n) = Set(apply(k -> bitxor(n-k, k), [0..n]))