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

0,19

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.

Links

Table of n, a(n) for n=0..90.

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

Example

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

Prog

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

Crossrefs

See A369312 for the corresponding quotients.

Cf. A048158, A048720.

Sequence in context: A324904 A109708 A035468 * A263860 A051777 A262709

Adjacent sequences: A369308 A369309 A369310 * A369312 A369313 A369314

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Jan 19 2024

Status

approved