A341318 Lower triangular table of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition), read by rows.

0, 0, 1, 0, 2, 7, 0, 3, 8, 10, 0, 4, 16, 17, 37, 0, 5, 17, 19, 38, 40, 0, 6, 18, 21, 39, 42, 45, 0, 7, 29, 30, 67, 68, 69, 121, 0, 8, 30, 32, 68, 70, 72, 122, 124, 0, 9, 31, 34, 69, 72, 75, 123, 126, 129, 0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136, 0, 11, 46, 47, 106, 107, 108, 191, 192, 193, 194, 301

Offset: 0

Author

N. J. A. Sloane, Feb 17 2021

Keywords

• nonn
• tabl

Comments

Consider the semigroup S consisting of the pairs (0,0) and {(i,j): i >= j >= 1}, with componentwise products. Label the elements 0 = (0,0), 1 = (1,1), 2 = (2,1), 3 = (2,2), 4 = (3,1), 5 = (3,2), 6 = (3,3), 7 = (4,1), ... The triangle gives T(n,k) = label of product of n-th and k-th elements, for n>=k>=0.

See A341317 for further information, including a Maple program.

Examples

			Triangle begins:
0, [0]
1, [0, 1]
2, [0, 2, 7]
3, [0, 3, 8, 10]
4, [0, 4, 16, 17, 37]
5, [0, 5, 17, 19, 38, 40]
6, [0, 6, 18, 21, 39, 42, 45]
7, [0, 7, 29, 30, 67, 68, 69, 121]
8, [0, 8, 30, 32, 68, 70, 72, 122, 124]
9, [0, 9, 31, 34, 69, 72, 75, 123, 126, 129]
10, [0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136]
...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A341317, A341706.

Main diagonal gives A341736.

Programs

Maple

t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)][])(f(n)):
T:= (n, k)-> (h-> t(h[1]*h[3])+h[2]*h[4])(map(g, [n, k])):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 17 2021

Mathematica

t[n_] := n*(n - 1)/2;
f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
g[n_] := Function[x, {x, n - t[x]}][f[n]];
T[n_, k_] := (Function[h, t[h[[1]]*h[[3]]] + h[[2]]*h[[4]]])[Flatten @ Map[g, {n, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)