A342639 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = g(f(n) + f(k)) where g maps the complement, say s, of a finite set of nonnegative integers to the value Sum_{e >= 0 not in s} 2^e, f is the inverse of g, and "+" denotes the Minkowski sum.

0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 7, 2, 7, 0, 1, 1, 3, 3, 1, 1, 0, 3, 0, 15, 0, 3, 0, 7, 1, 5, 3, 3, 5, 1, 7, 0, 15, 2, 7, 0, 7, 2, 15, 0, 1, 1, 7, 3, 1, 1, 3, 7, 1, 1, 0, 3, 0, 31, 4, 11, 4, 31, 0, 3, 0, 3, 1, 1, 3, 7, 5, 5, 7, 3, 1, 1, 3, 0, 7, 2, 7, 0, 15, 6, 15, 0, 7, 2, 7, 0

Offset

0,5

Comments

In other words:

- we consider the set S of sets s of nonnegative integers whose complement is finite,

- the function g encodes the "missing integers" in binary:

g(A001477 \ {1, 4}) = 2^1 + 2^4 = 18

- the function f is the inverse of g:

f(42) = f(2^1 + 2^3 + 2^5) = A001477 \ {1, 3, 5},

- the Minkowski sum of two sets, say U and V, is the set of sums u+v where u belongs to U and v belongs to V,

- the Minkowski sum is stable over S,

- and T provides an encoding for this operation.

This sequence has connections with A067138; here we consider complements of finite sets of nonnegative integers, there finite sets of nonnegative integers.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Wikipedia, Minkowski addition

Formula

T(n, k) = T(k, n).

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

T(n, 0) = A135481(n).

T(n, 1) = A038712(n+1).

T(2^n-1, 2^k-1) = 2^(n+k)-1.

T(n, n) = A342640(n).

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   1   0   3   0   1   0    7   0   1   0   3   0   1   0   15
    1|   1   3   1   7   1   3   1   15   1   3   1   7   1   3   1   31
    2|   0   1   2   3   0   5   2    7   0   1   2  11   0   5   2   15
    3|   3   7   3  15   3   7   3   31   3   7   3  15   3   7   3   63
    4|   0   1   0   3   0   1   4    7   0   1   0   3   0   9   4   15
    5|   1   3   5   7   1  11   5   15   1   3   5  23   1  11   5   31
    6|   0   1   2   3   4   5   6    7   0   9   2  11   4  13   6   15
    7|   7  15   7  31   7  15   7   63   7  15   7  31   7  15   7  127
    8|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   8   15
    9|   1   3   1   7   1   3   9   15   1   3   1   7   1  19   9   31
   10|   0   1   2   3   0   5   2    7   0   1  10  11   0   5  10   15
   11|   3   7  11  15   3  23  11   31   3   7  11  47   3  23  11   63
   12|   0   1   0   3   0   1   4    7   0   1   0   3   8   9  12   15
   13|   1   3   5   7   9  11  13   15   1  19   5  23   9  27  13   31
   14|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
   15|  15  31  15  63  15  31  15  127  15  31  15  63  15  31  15  255

Prog

(PARI) T(n, k) = { my (v=0); for (x=0, #binary(n)+#binary(k), my (f=0); for (y=0, x, if (!bittest(n, y) && !bittest(k, x-y), f=1; break)); if (!f, v+=2^x)); return (v) }

Crossrefs

Cf. A038712, A067138, A133457, A135481, A342640, A342641, A342642.

Sequence in context: A350950 A051171 A336118 * A300904 A307807 A350849

Adjacent sequences: A342636 A342637 A342638 * A342640 A342641 A342642

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Mar 17 2021

Status

approved