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A322744 Array T(n,k) = (3*n*k - A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

%I A322744 #47 Jul 10 2022 20:43:14
%S A322744 1,2,2,3,6,3,4,8,8,4,5,12,11,12,5,6,14,16,16,14,6,7,18,19,24,19,18,7,
%T A322744 8,20,24,28,28,24,20,8,9,24,27,36,33,36,27,24,9,10,26,32,40,42,42,40,
%U A322744 32,26,10,11,30,35,48,47,54,47,48,35,30,11,12,32,40,52,56,60,60,56,52,40,32,12
%N A322744 Array T(n,k) = (3*n*k - A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.
%C A322744 Associative multiplication-like table whose values depend on whether n and k are odd or even.
%C A322744 Associativity is proved by checking the formula with eight cases of three odd and even arguments. T(n,k) is distributive as long as partitioning an even number into two odd numbers is not allowed.
%H A322744 David Lovler, <a href="/A322744/b322744.txt">Table of n, a(n) for n = 1..861</a> (Antidiagonals n = 1..41, flattened)
%F A322744 T(n,k) = (3*n*k - (n + k - 1))/2 if n is odd and k is odd;
%F A322744 T(n,k) = (3*n*k - n)/2 if n is even and k is odd;
%F A322744 T(n,k) = (3*n*k - k)/2 if n is odd and k is even;
%F A322744 T(n,k) = 3*n*k/2 if n is even and k is even.
%F A322744 T(n,k) = 6*floor(n/2)*floor(k/2) + A319929(n,k).
%F A322744 T(n,n) = A354594(n). - _David Lovler_, Jul 09 2022
%e A322744 Array T(n,k) begins:
%e A322744    1   2   3   4   5   6   7   8   9  10
%e A322744    2   6   8  12  14  18  20  24  26  30
%e A322744    3   8  11  16  19  24  27  32  35  40
%e A322744    4  12  16  24  28  36  40  48  52  60
%e A322744    5  14  19  28  33  42  47  56  61  70
%e A322744    6  18  24  36  42  54  60  72  78  90
%e A322744    7  20  27  40  47  60  67  80  87 100
%e A322744    8  24  32  48  56  72  80  96 104 120
%e A322744    9  26  35  52  61  78  87 104 113 130
%e A322744   10  30  40  60  70  90 100 120 130 150
%t A322744 Table[Function[n, (3 n k - If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]])/2][m - k + 1], {m, 12}, {k, m}] // Flatten (* _Michael De Vlieger_, Apr 21 2019 *)
%o A322744 (PARI) T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
%o A322744 T(n,k) = (3*n*k - T319929(n,k))/2;
%o A322744 matrix(6, 6, n, k, T(n, k)) \\ _Michel Marcus_, Dec 27 2018
%Y A322744 Equals A003991 + A322630 - A319929.
%Y A322744 Cf. A327263, A307001, A307002, A340746, A340747.
%Y A322744 0 and diagonal is A354594.
%K A322744 nonn,tabl,easy
%O A322744 1,2
%A A322744 _David Lovler_, Dec 24 2018