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

0,4

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.

Links

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

Example

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;
    ...

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 *)

Prog

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

Crossrefs

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

Cf. A002487, A135481, A328565.

Sequence in context: A024307 A369287 A267852 * A219864 A257844 A194745

Adjacent sequences: A328565 A328566 A328567 * A328569 A328570 A328571

Keyword

nonn,tabf,look,base

Author

Rémy Sigrist, Oct 20 2019

Status

approved