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%I #30 Aug 03 2020 00:22:21
%S 1,2,2,3,3,3,4,5,5,4,5,6,8,6,5,6,8,10,10,8,6,7,9,13,12,13,9,7,8,11,15,
%T 16,16,15,11,8,9,12,18,18,21,18,18,12,9,10,14,20,22,24,24,22,20,14,10,
%U 11,15,23,24,29,27,29,24,23,15,11
%N Square array T(n,k) = ((3/2)*n*k + (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.
%C 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.
%C T(n,k) is in the same form as the supplementary arrays of A327263 called U(i;n,k). Here (and in A334923) 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.
%H David Lovler, <a href="/A334922/b334922.txt">Table of n, a(n) for n = 1..465</a>
%F T(n,k) = 3*floor(n/2)*floor(k/2) + A319929(n,k).
%F T(n,k) = (A322630(n,k) + n*k)/2.
%F T(n,k) = (A319929 + A322744(n,k))/2.
%F T(n,k) = 2*n*k - A334923(n,k).
%e Array begins:
%e 1 2 3 4 5 6 7 8 9 10 ...
%e 2 3 5 6 8 9 11 12 14 15 ...
%e 3 5 8 10 13 15 18 20 23 25 ...
%e 4 6 10 12 16 18 22 24 28 30 ...
%e 5 8 13 16 21 24 29 32 37 40 ...
%e 6 9 15 18 24 27 33 36 42 45 ...
%e 7 11 18 22 29 33 40 44 51 55 ...
%e 8 12 20 24 32 36 44 48 56 60 ...
%e 9 14 23 28 37 42 51 56 65 70 ...
%e 10 15 25 30 40 45 55 60 70 75 ...
%e ...
%t Table[Function[n, ((3/2)*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 *)
%Y Cf. A319929, A322630, A322744, A327259, A327263, A334923.
%K nonn,tabl
%O 1,2
%A _David Lovler_, May 16 2020
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