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%I #12 Feb 24 2021 08:20:45
%S 0,1,0,2,0,0,3,3,0,0,4,3,0,1,0,5,4,0,1,1,0,6,4,7,0,1,0,0,7,7,7,7,0,0,
%T 0,0,8,7,7,6,0,0,3,1,0,9,8,7,6,1,0,3,2,1,0,10,8,8,7,1,0,3,3,2,0,0,11,
%U 11,8,8,0,0,3,3,3,3,0,0,12,11,8,9,15,0,0,2,3,3,0,1,0
%N Square array T(n, k), n, k >= 0, read by antidiagonals upwards; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) equals R(n) minus R(k).
%C For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
%C This sequence uses set subtraction, and is related to:
%C - A003987 which uses set difference,
%C - A341839 which uses set union,
%C - A341840 which uses set intersection.
%H Rémy Sigrist, <a href="/A341841/b341841.txt">Table of n, a(n) for n = 0..10010</a>
%H Rémy Sigrist, <a href="/A341841/a341841.png">Colored representation of the table for n, k < 2^10</a>
%H Rémy Sigrist, <a href="/A341841/a341841.gp.txt">PARI program for A341841</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F T(n, n) = 0.
%F T(n, 0) = n.
%F T(T(n, k), k) = T(n, k).
%F A070939(T(n, k)) <= A070939(n).
%F A003188(T(n, k)) = A003188(n) - (A003188(n) AND A003188(k)) (where AND denotes the bitwise AND operator).
%e Array T(n, k) begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e ---+--------------------------------------------------------------
%e 0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 1| 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1
%e 2| 2 3 0 1 1 0 3 2 2 3 0 1 1 0 3 2
%e 3| 3 3 0 0 0 0 3 3 3 3 0 0 0 0 3 3
%e 4| 4 4 7 7 0 0 3 3 3 3 0 0 7 7 4 4
%e 5| 5 4 7 6 1 0 3 2 2 3 0 1 6 7 4 5
%e 6| 6 7 7 6 1 0 0 1 1 0 0 1 6 7 7 6
%e 7| 7 7 7 7 0 0 0 0 0 0 0 0 7 7 7 7
%e 8| 8 8 8 8 15 15 15 15 0 0 0 0 7 7 7 7
%e 9| 9 8 8 9 14 15 15 14 1 0 0 1 6 7 7 6
%e 10| 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5
%e 11| 11 11 8 8 15 15 12 12 3 3 0 0 7 7 4 4
%e 12| 12 12 15 15 15 15 12 12 3 3 0 0 0 0 3 3
%e 13| 13 12 15 14 14 15 12 13 2 3 0 1 1 0 3 2
%e 14| 14 15 15 14 14 15 15 14 1 0 0 1 1 0 0 1
%e 15| 15 15 15 15 15 15 15 15 0 0 0 0 0 0 0 0
%o (PARI) See Links section.
%Y Cf. A003188, A003987, A070939, A227736, A341839, A341840.
%K nonn,tabl,base
%O 0,4
%A _Rémy Sigrist_, Feb 21 2021