A322744 Array T(n,k) = (3*n*k - A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.

1, 2, 2, 3, 6, 3, 4, 8, 8, 4, 5, 12, 11, 12, 5, 6, 14, 16, 16, 14, 6, 7, 18, 19, 24, 19, 18, 7, 8, 20, 24, 28, 28, 24, 20, 8, 9, 24, 27, 36, 33, 36, 27, 24, 9, 10, 26, 32, 40, 42, 42, 40, 32, 26, 10, 11, 30, 35, 48, 47, 54, 47, 48, 35, 30, 11, 12, 32, 40, 52, 56, 60, 60, 56, 52, 40, 32, 12

Offset: 1

Author

David Lovler, Dec 24 2018

Keywords

• nonn
• tabl
• easy

Comments

Associative multiplication-like table whose values depend on whether n and k are odd or even.

Associativity is proved by checking the formula with eight cases of three odd and even arguments. T(n,k) is distributive as long as partitioning an even number into two odd numbers is not allowed.

Examples

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   6   8  12  14  18  20  24  26  30
   3   8  11  16  19  24  27  32  35  40
   4  12  16  24  28  36  40  48  52  60
   5  14  19  28  33  42  47  56  61  70
   6  18  24  36  42  54  60  72  78  90
   7  20  27  40  47  60  67  80  87 100
   8  24  32  48  56  72  80  96 104 120
   9  26  35  52  61  78  87 104 113 130
  10  30  40  60  70  90 100 120 130 150
		

Links

• David Lovler, Table of n, a(n) for n = 1..861 (Antidiagonals n = 1..41, flattened)

Crossrefs

Equals A003991 + A322630 - A319929.

Cf. A327263, A307001, A307002, A340746, A340747.

0 and diagonal is A354594.

Programs

Mathematica

Table[Function[n, (3 n k - If[OddQ@ n, If[OddQ@ k, n + k - 1, k], If[OddQ@ k, n, 0]])/2][m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Apr 21 2019 *)

PARI

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n,k) = (3*n*k - T319929(n,k))/2;
matrix(6, 6, n, k, T(n, k)) \\ Michel Marcus, Dec 27 2018

Formula

T(n,k) = (3*n*k - (n + k - 1))/2 if n is odd and k is odd;

T(n,k) = (3*n*k - n)/2 if n is even and k is odd;

T(n,k) = (3*n*k - k)/2 if n is odd and k is even;

T(n,k) = 3*n*k/2 if n is even and k is even.

T(n,k) = 6*floor(n/2)*floor(k/2) + A319929(n,k).

T(n,n) = A354594(n). - David Lovler, Jul 09 2022