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

1,2

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.

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Equals A003991 + A322630 - A319929.

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

0 and diagonal is A354594.

Sequence in context: A291372 A064426 A051173 * A128228 A190098 A272973

Adjacent sequences: A322741 A322742 A322743 * A322745 A322746 A322747

Keyword

nonn,tabl,easy

Author

David Lovler, Dec 24 2018

Status

approved