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

0,4

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.

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 |  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

Prog

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

Crossrefs

See A369311 for the corresponding remainders.

Cf. A048720, A010766.

Sequence in context: A288437 A287736 A180969 * A259479 A238343 A238128

Adjacent sequences: A369309 A369310 A369311 * A369313 A369314 A369315

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Jan 19 2024

Status

approved