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A334923
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Square array T(n,k) = ((5/2)*n*k - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.
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2
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1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 10, 10, 10, 5, 6, 12, 14, 14, 12, 6, 7, 15, 17, 20, 17, 15, 7, 8, 17, 21, 24, 24, 21, 17, 8, 9, 20, 24, 30, 29, 30, 24, 20, 9, 10, 22, 28, 34, 36, 36, 34, 28, 22, 10, 11, 25, 31, 40, 41, 45, 41, 40, 31, 25, 11
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OFFSET
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1,2
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COMMENTS
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T(n,k) is commutative, associative, has identity element 1 and has 0. Also it is distributive except when an even number is partitioned into two odd numbers. Thus it has a multiplicative structure similar to that of A319929, A322630, A322744 and A327259 except that T(odd,odd) is not always odd, T(even,even) is not always even and T(odd,even) is not always even.
T(n,k) is in the same form as the supplementary arrays of A327263 called U(i;n,k). Here (and in A334922) i is being incremented by 1/2. When i is incremented by 1/4 or less, array values cease to be all integers, although all of the multiplication rules still hold.
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LINKS
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FORMULA
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T(n,k) = 5*floor(n/2)*floor(k/2) + A319929(n,k).
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EXAMPLE
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Array begins:
1 2 3 4 5 6 7 8 9 10 ...
2 5 7 10 12 15 17 20 22 25 ...
3 7 10 14 17 21 24 28 31 35 ...
4 10 14 20 24 30 34 40 44 50 ...
5 12 17 24 29 36 41 48 53 60 ...
6 15 21 30 36 45 51 60 66 75 ...
7 17 24 34 41 51 58 68 75 85 ...
8 20 28 40 48 60 68 80 88 100 ...
9 22 31 44 53 66 75 88 97 110 ...
10 25 35 50 60 75 85 100 110 125 ...
...
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MATHEMATICA
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Table[Function[n, ((5/2)*n*k - (1/2)*If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]])/2][m - k + 1], {m, 11}, {k, m}] // Flatten (* Michael De Vlieger, Jun 23 2020 *)
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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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