A369312 - OEIS

%I #6 Jan 21 2024 09:37:44

%S 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,

%T 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,

%U 5,7,2,1,1,1,0,0,0,0,0,12,5,6,2,2,1,1,0,0,0,0,0,0

%N 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).

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

%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>

%e Array A(n, k) begins:

%e   n\k |  1  2  3  4  5  6  7  8  9  10  11  12

%e   ----+---------------------------------------

%e     0 |  0  0  0  0  0  0  0  0  0   0   0   0

%e     1 |  1  0  0  0  0  0  0  0  0   0   0   0

%e     2 |  2  1  1  0  0  0  0  0  0   0   0   0

%e     3 |  3  1  1  0  0  0  0  0  0   0   0   0

%e     4 |  4  2  3  1  1  1  1  0  0   0   0   0

%e     5 |  5  2  3  1  1  1  1  0  0   0   0   0

%e     6 |  6  3  2  1  1  1  1  0  0   0   0   0

%e     7 |  7  3  2  1  1  1  1  0  0   0   0   0

%e     8 |  8  4  7  2  2  3  3  1  1   1   1   1

%e     9 |  9  4  7  2  2  3  3  1  1   1   1   1

%e    10 | 10  5  6  2  2  3  3  1  1   1   1   1

%e    11 | 11  5  6  2  2  3  3  1  1   1   1   1

%e    12 | 12  6  4  3  3  2  2  1  1   1   1   1

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

%Y See A369311 for the corresponding remainders.

%Y Cf. A048720, A010766.

%K nonn,base,tabl

%O 0,4

%A _Rémy Sigrist_, Jan 19 2024