A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 7, 4, 5, 6, 6, 8, 8, 6, 6, 7, 6, 11, 8, 11, 6, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 8, 15, 12, 17, 12, 15, 8, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 10, 19, 16, 23, 18, 23, 16, 19, 10, 11

Offset: 1

Author

David Lovler, Dec 20 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.

Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j > 0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.

Examples

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   2   4   4   6   6   8   8  10  10
   3   4   7   8  11  12  15  16  19  20
   4   4   8   8  12  12  16  16  20  20
   5   6  11  12  17  18  23  24  29  30
   6   6  12  12  18  18  24  24  30  30
   7   8  15  16  23  24  31  32  39  40
   8   8  16  16  24  24  32  32  40  40
   9  10  19  20  29  30  39  40  49  50
  10  10  20  20  30  30  40  40  50  50
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened).

Crossrefs

Cf. A076274, A327263.

0 and diagonal is A213037.

Programs

Mathematica

Table[Function[n, Switch[FromDigits[Mod[{n, k}, 2], 2], 0, n k/2, 1, (n k + n)/2, 2, (n k + k)/2, , (n k + n + k - 1)/2]][m - k + 1], {m, 11}, {k, m}] // Flatten (* _Michael De Vlieger, Jan 14 2022 *)

PARI

T319929(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n,k) = (T319929(n,k) + n*k)/2;
matrix(6, 6, n, k, T(n,k)) \\ Michel Marcus, Dec 22 2018

Formula

T(n,k) = (n*k + n + k - 1)/2 if n is odd and k is odd;

T(n,k) = (n*k + n)/2 if n is even and k is odd;

T(n,k) = (n*k + k)/2 if n is odd and k is even;

T(n,k) = n*k/2 if n is even and k is even.

Extensions

Name clarified by David Lovler, Jan 24 2022