A369312 Square array A(n, k), n >= 0, k > 0, read by upwards antidiagonals: P(A(n, k)) is the quotient of the polynomial long division of P(n) by P(k) (where P(m) denotes the polynomial over GF(2) whose coefficients are encoded in the binary expansion of the nonnegative integer m).

0, 1, 0, 2, 0, 0, 3, 1, 0, 0, 4, 1, 1, 0, 0, 5, 2, 1, 0, 0, 0, 6, 2, 3, 0, 0, 0, 0, 7, 3, 3, 1, 0, 0, 0, 0, 8, 3, 2, 1, 1, 0, 0, 0, 0, 9, 4, 2, 1, 1, 1, 0, 0, 0, 0, 10, 4, 7, 1, 1, 1, 1, 0, 0, 0, 0, 11, 5, 7, 2, 1, 1, 1, 0, 0, 0, 0, 0, 12, 5, 6, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Rémy Sigrist, Jan 19 2024

Keywords

• nonn
• base
• tabl

Comments

For any number m >= 0 with binary expansion Sum_{k >= 0} b_k * 2^k, P(m) = Sum_{k >= 0} b_k * X^k.

Examples

			Array A(n, k) begins:
  n\k |  1  2  3  4  5  6  7  8  9  10  11  12
  ----+---------------------------------------
    0 |  0  0  0  0  0  0  0  0  0   0   0   0
    1 |  1  0  0  0  0  0  0  0  0   0   0   0
    2 |  2  1  1  0  0  0  0  0  0   0   0   0
    3 |  3  1  1  0  0  0  0  0  0   0   0   0
    4 |  4  2  3  1  1  1  1  0  0   0   0   0
    5 |  5  2  3  1  1  1  1  0  0   0   0   0
    6 |  6  3  2  1  1  1  1  0  0   0   0   0
    7 |  7  3  2  1  1  1  1  0  0   0   0   0
    8 |  8  4  7  2  2  3  3  1  1   1   1   1
    9 |  9  4  7  2  2  3  3  1  1   1   1   1
   10 | 10  5  6  2  2  3  3  1  1   1   1   1
   11 | 11  5  6  2  2  3  3  1  1   1   1   1
   12 | 12  6  4  3  3  2  2  1  1   1   1   1
		

Links

• Index entries for sequences operating on GF(2)[X]-polynomials

Crossrefs

See A369311 for the corresponding remainders.

Cf. A048720, A010766.

Programs

PARI

A(n, k) = { fromdigits(lift(Vec( (Mod(1, 2) * Pol(binary(n))) \ (Mod(1, 2) * Pol(binary(k))))), 2) }