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Minimal arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals.
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%I #87 Feb 21 2022 10:52:38

%S 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 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 11,12,2,10,4,8,6,6,8,4,10,2,12

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

%C 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 Michael De Vlieger, <a href="/A319929/b319929.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)

%H 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 David Lovler, <a href="/A319929/a319929_2.pdf">Motivation</a>

%F T(n,k) = n + k - 1 if n is odd and k is odd;

%F T(n,k) = n if n is even and k is odd;

%F T(n,k) = k if n is odd and k is even;

%F T(n,k) = 0 if n is even and k is even.

%e T(3,5) = 3 + 5 - 1 = 7, T(4,7) = 4, T(8,8) = 0.

%e Array T(n,k) begins:

%e 1 2 3 4 5 6 7 8 9 10

%e 2 0 2 0 2 0 2 0 2 0

%e 3 2 5 4 7 6 9 8 11 10

%e 4 0 4 0 4 0 4 0 4 0

%e 5 2 7 4 9 6 11 8 13 10

%e 6 0 6 0 6 0 6 0 6 0

%e 7 2 9 4 11 6 13 8 15 10

%e 8 0 8 0 8 0 8 0 8 0

%e 9 2 11 4 13 6 15 8 17 10

%e 10 0 10 0 10 0 10 0 10 0

%t 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 (PARI) T(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));

%o matrix(6, 6, n, k, T(n,k)) \\ _Michel Marcus_, Dec 22 2018

%Y Cf. A322630, A322744, A327259, A327263.

%K nonn,tabl,easy

%O 1,2

%A _David Lovler_, Dec 17 2018