

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.


4



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
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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 nth and kth 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*(n1)/2:
f:= n> ceil((sqrt(1+8*n)1)/2):
g:= n> (x> [x, nt(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


CROSSREFS

Cf. A341317, A341706.
Main diagonal gives A341736.
Sequence in context: A188737 A200680 A260129 * A332324 A101689 A175292
Adjacent sequences: A341315 A341316 A341317 * A341319 A341320 A341321


KEYWORD

nonn,tabl


AUTHOR

N. J. A. Sloane, Feb 17 2021


STATUS

approved



