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%I #9 Jan 21 2024 09:38:05
%S 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,
%T 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,
%U 0,0,2,0,2,4,3,2,1,0,0,1,0,1,2,1,1,5,4,3,2,1,0
%N 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).
%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 Rémy Sigrist, <a href="/A369311/a369311.png">Colored scatterplot of (n, k) for n, k < 2^10</a> (where the color is function of A(n, 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 | 0 1 1 1 1 1 1 1 1 1 1 1
%e 2 | 0 0 1 2 2 2 2 2 2 2 2 2
%e 3 | 0 1 0 3 3 3 3 3 3 3 3 3
%e 4 | 0 0 1 0 1 2 3 4 4 4 4 4
%e 5 | 0 1 0 1 0 3 2 5 5 5 5 5
%e 6 | 0 0 0 2 3 0 1 6 6 6 6 6
%e 7 | 0 1 1 3 2 1 0 7 7 7 7 7
%e 8 | 0 0 1 0 2 2 1 0 1 2 3 4
%e 9 | 0 1 0 1 3 3 0 1 0 3 2 5
%e 10 | 0 0 0 2 0 0 3 2 3 0 1 6
%e 11 | 0 1 1 3 1 1 2 3 2 1 0 7
%e 12 | 0 0 0 0 3 0 2 4 5 6 7 0
%o (PARI) A(n, k) = { fromdigits(lift(Vec( (Mod(1, 2) * Pol(binary(n))) % (Mod(1, 2) * Pol(binary(k))))), 2) }
%Y See A369312 for the corresponding quotients.
%Y Cf. A048158, A048720.
%K nonn,base,tabl
%O 0,19
%A _Rémy Sigrist_, Jan 19 2024