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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.

%I A341318 #21 Feb 26 2022 10:45:38
%S A341318 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,
%T A341318 0,7,29,30,67,68,69,121,0,8,30,32,68,70,72,122,124,0,9,31,34,69,72,75,
%U A341318 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
%N 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.
%C A341318 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.
%C A341318 See A341317 for further information, including a Maple program.
%H A341318 Alois P. Heinz, <a href="/A341318/b341318.txt">Rows n = 0..200, flattened</a>
%e A341318 Triangle begins:
%e A341318 0, [0]
%e A341318 1, [0, 1]
%e A341318 2, [0, 2, 7]
%e A341318 3, [0, 3, 8, 10]
%e A341318 4, [0, 4, 16, 17, 37]
%e A341318 5, [0, 5, 17, 19, 38, 40]
%e A341318 6, [0, 6, 18, 21, 39, 42, 45]
%e A341318 7, [0, 7, 29, 30, 67, 68, 69, 121]
%e A341318 8, [0, 8, 30, 32, 68, 70, 72, 122, 124]
%e A341318 9, [0, 9, 31, 34, 69, 72, 75, 123, 126, 129]
%e A341318 10, [0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136]
%e A341318 ...
%p A341318 t:= n-> n*(n-1)/2:
%p A341318 f:= n-> ceil((sqrt(1+8*n)-1)/2):
%p A341318 g:= n-> (x-> [x, n-t(x)][])(f(n)):
%p A341318 T:= (n, k)-> (h-> t(h[1]*h[3])+h[2]*h[4])(map(g, [n, k])):
%p A341318 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Feb 17 2021
%t A341318 t[n_] := n*(n - 1)/2;
%t A341318 f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
%t A341318 g[n_] := Function[x, {x, n - t[x]}][f[n]];
%t A341318 T[n_, k_] := (Function[h, t[h[[1]]*h[[3]]] + h[[2]]*h[[4]]])[Flatten @ Map[g, {n, k}]];
%t A341318 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 26 2022, after _Alois P. Heinz_ *)
%Y A341318 Cf. A341317, A341706.
%Y A341318 Main diagonal gives A341736.
%K A341318 nonn,tabl
%O A341318 0,5
%A A341318 _N. J. A. Sloane_, Feb 17 2021