login
A322744
Array T(n,k) = (3*n*k - A319929(n,k))/2, n >= 1, k >= 1, read by antidiagonals.
13
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.
0 and diagonal is A354594.
Sequence in context: A291372 A064426 A051173 * A128228 A190098 A272973
KEYWORD
nonn,tabl,easy
AUTHOR
David Lovler, Dec 24 2018
STATUS
approved