A352727 Square array A(n, k), n, k >= 0, read by antidiagonals: the binary expansion of A(n, k) contains the runs of consecutive 1's that appear both in the binary expansions of n and k.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0

Offset: 0

Author

Rémy Sigrist, Mar 30 2022

Keywords

• nonn
• base
• tabl

Comments

We only consider maximal runs of one or more consecutive 1's (as counted by A069010) that completely match in binary expansions of n and k, not simply single common 1's.

Examples

			Table A(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  0  0  0  1  0  0  0  1   0   0   0   1   0   0
    2|  0  0  2  0  0  0  0  0  0  0   2   0   0   0   0   0
    3|  0  0  0  3  0  0  0  0  0  0   0   3   0   0   0   0
    4|  0  0  0  0  4  4  0  0  0  0   0   0   0   0   0   0
    5|  0  1  0  0  4  5  0  0  0  1   0   0   0   1   0   0
    6|  0  0  0  0  0  0  6  0  0  0   0   0   0   0   0   0
    7|  0  0  0  0  0  0  0  7  0  0   0   0   0   0   0   0
    8|  0  0  0  0  0  0  0  0  8  8   8   8   0   0   0   0
    9|  0  1  0  0  0  1  0  0  8  9   8   8   0   1   0   0
   10|  0  0  2  0  0  0  0  0  8  8  10   8   0   0   0   0
   11|  0  0  0  3  0  0  0  0  8  8   8  11   0   0   0   0
   12|  0  0  0  0  0  0  0  0  0  0   0   0  12  12   0   0
   13|  0  1  0  0  0  1  0  0  0  1   0   0  12  13   0   0
   14|  0  0  0  0  0  0  0  0  0  0   0   0   0   0  14   0
   15|  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0  15
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k); black pixels denote 0's)
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A004198, A069010, A352724, A352729.

Programs

PARI

A352724(n) = { my (r=[], o=0); while (n, my (v=valuation(n+n%2, 2)); if (n%2, r=concat(r, (2^v-1)*2^o)); o+=v; n\=2^v); r }
A(n,k) = vecsum(setintersect(A352724(n), A352724(k)))

Formula

A(n, k) = A(k, n).

A(n, 0) = 0.

A(n, n) = n.

A(n, 2*n) = 0.

A(n, k) <= A004198(n, k) (bitwise AND operator).

A(n, n+1) = A352729(n).