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Square array T(n, k), n, k >= 0, read by antidiagonals; 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) is the union of R(n) and of R(k).
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%I #17 Feb 24 2021 08:20:29

%S 0,1,1,2,1,2,3,2,2,3,4,2,2,2,4,5,5,2,2,5,5,6,5,5,3,5,5,6,7,6,5,4,4,5,

%T 6,7,8,6,5,5,4,5,5,6,8,9,9,5,5,5,5,5,5,9,9,10,9,10,4,5,5,5,4,10,9,10,

%U 11,10,10,11,4,5,5,4,11,10,10,11,12,10,10,10,11,5,6,5,11,10,10,10,12

%N Square array T(n, k), n, k >= 0, read by antidiagonals; 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) is the union of R(n) and of 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 The underlying idea is to break in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.

%H Rémy Sigrist, <a href="/A341839/b341839.txt">Table of n, a(n) for n = 0..10010</a>

%H Rémy Sigrist, <a href="/A341839/a341839.png">Colored representation of the table for n, k < 2^10</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F T(n, k) = T(k, n)

%F T(m, T(n, k)) = T(T(m, n), k).

%F T(n, n) = n.

%F T(n, 0) = 0.

%F A070939(T(n, k)) = max(A070939(n), A070939(k)).

%F A003188(T(n, k)) = A003188(n) OR A003188(k) (where OR denotes the bitwise OR operator).

%F T(n, 1) = A042963(ceiling((n+1)/2)).

%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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

%e 1| 1 1 2 2 5 5 6 6 9 9 10 10 13 13 14 14

%e 2| 2 2 2 2 5 5 5 5 10 10 10 10 13 13 13 13

%e 3| 3 2 2 3 4 5 5 4 11 10 10 11 12 13 13 12

%e 4| 4 5 5 4 4 5 5 4 11 10 10 11 11 10 10 11

%e 5| 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10

%e 6| 6 6 5 5 5 5 6 6 9 9 10 10 10 10 9 9

%e 7| 7 6 5 4 4 5 6 7 8 9 10 11 11 10 9 8

%e 8| 8 9 10 11 11 10 9 8 8 9 10 11 11 10 9 8

%e 9| 9 9 10 10 10 10 9 9 9 9 10 10 10 10 9 9

%e 10| 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

%e 11| 11 10 10 11 11 10 10 11 11 10 10 11 11 10 10 11

%e 12| 12 13 13 12 11 10 10 11 11 10 10 11 12 13 13 12

%e 13| 13 13 13 13 10 10 10 10 10 10 10 10 13 13 13 13

%e 14| 14 14 13 13 10 10 9 9 9 9 10 10 13 13 14 14

%e 15| 15 14 13 12 11 10 9 8 8 9 10 11 12 13 14 15

%o (PARI) T(n,k) = { my (r=[], v=0); while (n||k, my (w=min(valuation(n+n%2,2), valuation(k+k%2,2))); r=concat(w,r); n\=2^w; k\=2^w); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }

%Y Cf. A003188, A003987, A005811, A042963, A070939, A101211, A227736, A341840, A341841.

%K nonn,base,tabl

%O 0,4

%A _Rémy Sigrist_, Feb 21 2021