A319929 - OEIS

%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