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A319929 Minimal arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals.

%I A319929 #87 Feb 21 2022 10:52:38
%S A319929 1,2,2,3,0,3,4,2,2,4,5,0,5,0,5,6,2,4,4,2,6,7,0,7,0,7,0,7,8,2,6,4,4,6,
%T A319929 2,8,9,0,9,0,9,0,9,0,9,10,2,8,4,6,6,4,8,2,10,11,0,11,0,11,0,11,0,11,0,
%U A319929 11,12,2,10,4,8,6,6,8,4,10,2,12
%N A319929 Minimal arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals.
%C A319929 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.
%H A319929 Michael De Vlieger, <a href="/A319929/b319929.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%H A319929 Michael De Vlieger, <a href="/A319929/a319929.png">Array plot of T(n,k)</a> for n = 1..150, k = 1..150 with color function indicating value, pale yellow = 0, red = 299.
%H A319929 David Lovler, <a href="/A319929/a319929_2.pdf">Motivation</a>
%F A319929 T(n,k) = n + k - 1 if n is odd and k is odd;
%F A319929 T(n,k) = n if n is even and k is odd;
%F A319929 T(n,k) = k if n is odd and k is even;
%F A319929 T(n,k) = 0 if n is even and k is even.
%e A319929 T(3,5) = 3 + 5 - 1 = 7, T(4,7) = 4, T(8,8) = 0.
%e A319929 Array T(n,k) begins:
%e A319929    1  2  3  4  5  6  7  8  9 10
%e A319929    2  0  2  0  2  0  2  0  2  0
%e A319929    3  2  5  4  7  6  9  8 11 10
%e A319929    4  0  4  0  4  0  4  0  4  0
%e A319929    5  2  7  4  9  6 11  8 13 10
%e A319929    6  0  6  0  6  0  6  0  6  0
%e A319929    7  2  9  4 11  6 13  8 15 10
%e A319929    8  0  8  0  8  0  8  0  8  0
%e A319929    9  2 11  4 13  6 15  8 17 10
%e A319929   10  0 10  0 10  0 10  0 10  0
%t A319929 Table[Function[n, If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]]][m - k + 1], {m, 12}, {k, m}] // Flatten (* _Michael De Vlieger_, Mar 24 2019 *)
%o A319929 (PARI) T(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
%o A319929 matrix(6, 6, n, k, T(n,k)) \\ _Michel Marcus_, Dec 22 2018
%Y A319929 Cf. A322630, A322744, A327259, A327263.
%K A319929 nonn,tabl,easy
%O A319929 1,2
%A A319929 _David Lovler_, Dec 17 2018