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

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

Offset: 1

Author

David Lovler, Dec 17 2018

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			T(3,5) = 3 + 5 - 1 = 7, T(4,7) = 4, T(8,8) = 0.
Array T(n,k) begins:
   1  2  3  4  5  6  7  8  9 10
   2  0  2  0  2  0  2  0  2  0
   3  2  5  4  7  6  9  8 11 10
   4  0  4  0  4  0  4  0  4  0
   5  2  7  4  9  6 11  8 13 10
   6  0  6  0  6  0  6  0  6  0
   7  2  9  4 11  6 13  8 15 10
   8  0  8  0  8  0  8  0  8  0
   9  2 11  4 13  6 15  8 17 10
  10  0 10  0 10  0 10  0 10  0
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
• Michael De Vlieger, Array plot of T(n,k) for n = 1..150, k = 1..150 with color function indicating value, pale yellow = 0, red = 299.
• David Lovler, Motivation

Crossrefs

Cf. A322630, A322744, A327259, A327263.

Programs

Mathematica

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 *)

PARI

T(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
matrix(6, 6, n, k, T(n,k)) \\ Michel Marcus, Dec 22 2018

Formula

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

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

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

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