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A342639
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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.
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4
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(2^n-1, 2^k-1) = 2^(n+k)-1.
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EXAMPLE
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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
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PROG
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(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) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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