A369311 Square array A(n, k), n >= 0, k > 0, read by upwards antidiagonals: P(A(n, k)) is the remainder 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, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 3, 2, 1, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 1, 0, 1, 1, 3, 2, 1, 0, 0, 0, 1, 2, 0, 2, 3, 2, 1, 0, 0, 1, 1, 3, 3, 3, 3, 3, 2, 1, 0, 0, 0, 0, 0, 2, 0, 2, 4, 3, 2, 1, 0, 0, 1, 0, 1, 2, 1, 1, 5, 4, 3, 2, 1, 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 | 0  1  1  1  1  1  1  1  1   1   1   1
    2 | 0  0  1  2  2  2  2  2  2   2   2   2
    3 | 0  1  0  3  3  3  3  3  3   3   3   3
    4 | 0  0  1  0  1  2  3  4  4   4   4   4
    5 | 0  1  0  1  0  3  2  5  5   5   5   5
    6 | 0  0  0  2  3  0  1  6  6   6   6   6
    7 | 0  1  1  3  2  1  0  7  7   7   7   7
    8 | 0  0  1  0  2  2  1  0  1   2   3   4
    9 | 0  1  0  1  3  3  0  1  0   3   2   5
   10 | 0  0  0  2  0  0  3  2  3   0   1   6
   11 | 0  1  1  3  1  1  2  3  2   1   0   7
   12 | 0  0  0  0  3  0  2  4  5   6   7   0
		

Links

• Rémy Sigrist, Colored scatterplot of (n, k) for n, k < 2^10 (where the color is function of A(n, k))
• Index entries for sequences operating on GF(2)[X]-polynomials

Crossrefs

See A369312 for the corresponding quotients.

Cf. A048158, A048720.

Programs

PARI

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