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A322630
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Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.
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10
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1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 7, 4, 5, 6, 6, 8, 8, 6, 6, 7, 6, 11, 8, 11, 6, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 8, 15, 12, 17, 12, 15, 8, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 10, 19, 16, 23, 18, 23, 16, 19, 10, 11
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OFFSET
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1,2
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COMMENTS
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This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.
Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j > 0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.
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LINKS
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FORMULA
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T(n,k) = (n*k + n + k - 1)/2 if n is odd and k is odd;
T(n,k) = (n*k + n)/2 if n is even and k is odd;
T(n,k) = (n*k + k)/2 if n is odd and k is even;
T(n,k) = n*k/2 if n is even and k is even.
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EXAMPLE
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Array T(n,k) begins:
1 2 3 4 5 6 7 8 9 10
2 2 4 4 6 6 8 8 10 10
3 4 7 8 11 12 15 16 19 20
4 4 8 8 12 12 16 16 20 20
5 6 11 12 17 18 23 24 29 30
6 6 12 12 18 18 24 24 30 30
7 8 15 16 23 24 31 32 39 40
8 8 16 16 24 24 32 32 40 40
9 10 19 20 29 30 39 40 49 50
10 10 20 20 30 30 40 40 50 50
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MATHEMATICA
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Table[Function[n, Switch[FromDigits[Mod[{n, k}, 2], 2], 0, n k/2, 1, (n k + n)/2, 2, (n k + k)/2, _, (n k + n + k - 1)/2]][m - k + 1], {m, 11}, {k, m}] // Flatten (* Michael De Vlieger, Jan 14 2022 *)
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PROG
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(PARI) T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n, k) = (T319929(n, k) + n*k)/2;
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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