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

0,5

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.

Links

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

Example

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]
...

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 *)

Crossrefs

Cf. A341317, A341706.

Main diagonal gives A341736.

Sequence in context: A379388 A260129 A350763 * A332324 A101689 A175292

Adjacent sequences: A341315 A341316 A341317 * A341319 A341320 A341321

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 17 2021

Status

approved