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A341840
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 intersection of R(n) and of R(k).
4
0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 0, 0, 0, 1, 3, 3, 3, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 0, 1, 3, 4, 3, 1, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 7, 5, 7, 0, 0, 1, 0, 0, 1, 1, 0, 7, 6, 6, 7, 0, 1, 1, 0, 0, 0, 2, 0, 7, 7, 6, 7, 7, 0, 2, 0, 0
OFFSET
0,13
COMMENTS
For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
The underlying idea is to merge in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.
FORMULA
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
A070939(T(n, k)) <= min(A070939(n), A070939(k)).
A003188(T(n, k)) = A003188(n) AND A003188(k) (where AND denotes the bitwise AND operator).
EXAMPLE
Array T(n, k) begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---+--------------------------------------------------------
0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1| 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0
2| 0 1 2 3 3 2 1 0 0 1 2 3 3 2 1 0
3| 0 0 3 3 3 3 0 0 0 0 3 3 3 3 0 0
4| 0 0 3 3 4 4 7 7 7 7 4 4 3 3 0 0
5| 0 1 2 3 4 5 6 7 7 6 5 4 3 2 1 0
6| 0 1 1 0 7 6 6 7 7 6 6 7 0 1 1 0
7| 0 0 0 0 7 7 7 7 7 7 7 7 0 0 0 0
8| 0 0 0 0 7 7 7 7 8 8 8 8 15 15 15 15
9| 0 1 1 0 7 6 6 7 8 9 9 8 15 14 14 15
10| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
11| 0 0 3 3 4 4 7 7 8 8 11 11 12 12 15 15
12| 0 0 3 3 3 3 0 0 15 15 12 12 12 12 15 15
13| 0 1 2 3 3 2 1 0 15 14 13 12 12 13 14 15
14| 0 1 1 0 0 1 1 0 15 14 14 15 15 14 14 15
15| 0 0 0 0 0 0 0 0 15 15 15 15 15 15 15 15
PROG
(PARI) See Links section.
KEYWORD
nonn,tabl,base
AUTHOR
Rémy Sigrist, Feb 21 2021
STATUS
approved