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A334923 Square array T(n,k) = ((5/2)*n*k - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

%I A334923 #24 Aug 03 2020 00:22:54
%S A334923 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,
%T A334923 8,17,21,24,24,21,17,8,9,20,24,30,29,30,24,20,9,10,22,28,34,36,36,34,
%U A334923 28,22,10,11,25,31,40,41,45,41,40,31,25,11
%N A334923 Square array T(n,k) = ((5/2)*n*k - (1/2)*A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.
%C A334923 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 A334923 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.
%H A334923 David Lovler, <a href="/A334923/b334923.txt">Table of n, a(n) for n = 1..465</a>
%F A334923 T(n,k) = 5*floor(n/2)*floor(k/2) + A319929(n,k).
%F A334923 T(n,k) = (n*k + A322744(n,k))/2.
%F A334923 T(n,k) = (A322630(n,k) + A327259(n,k))/2.
%F A334923 T(n,k) = 2*n*k - A334922(n,k).
%e A334923 Array begins:
%e A334923 1   2   3   4   5   6   7   8   9  10 ...
%e A334923 2   5   7  10  12  15  17  20  22  25 ...
%e A334923 3   7  10  14  17  21  24  28  31  35 ...
%e A334923 4  10  14  20  24  30  34  40  44  50 ...
%e A334923 5  12  17  24  29  36  41  48  53  60 ...
%e A334923 6  15  21  30  36  45  51  60  66  75 ...
%e A334923 7  17  24  34  41  51  58  68  75  85 ...
%e A334923 8  20  28  40  48  60  68  80  88 100 ...
%e A334923 9  22  31  44  53  66  75  88  97 110 ...
%e A334923 10 25  35  50  60  75  85 100 110 125 ...
%e A334923 ...
%t A334923 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 *)
%Y A334923 Cf. A319929, A322630, A322744, A327259, A327263, A334922.
%K A334923 nonn,tabl
%O A334923 1,2
%A A334923 _David Lovler_, May 25 2020