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

1,2

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.

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

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A322630, A322744, A327259, A327263.

Sequence in context: A318448 A103516 A233558 * A129234 A213081 A127446

Adjacent sequences: A319926 A319927 A319928 * A319930 A319931 A319932

Keyword

nonn,tabl,easy

Author

David Lovler, Dec 17 2018

Status

approved