A341736 a(n) is the label of the square of the n-th element in the semigroup S = {(0,0), (i,j): i >= j >= 1}.

0, 1, 7, 10, 37, 40, 45, 121, 124, 129, 136, 301, 304, 309, 316, 325, 631, 634, 639, 646, 655, 666, 1177, 1180, 1185, 1192, 1201, 1212, 1225, 2017, 2020, 2025, 2032, 2041, 2052, 2065, 2080, 3241, 3244, 3249, 3256, 3265, 3276, 3289, 3304, 3321, 4951, 4954, 4959

Offset: 0

Author

Alois P. Heinz, Feb 17 2021

Keywords

• nonn

Comments

The product in S is computed componentwise.

For the labeling of the elements in S and further information see A341317.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Main diagonal of A341317 and of A341318.

Cf. A000217, A000290, A037270, A123578, A161680.

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)):
a:= n-> (h-> t(h[1]^2)+h[2]^2)(g(n)):
seq(a(n), n=0..60);

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]];
a[n_] := Function[h, t[h[[1]]^2] + h[[2]]^2][g[n]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

Formula

a(n) = A341317(n,n) = A341318(n,n).

a(A000217(n)) = A037270(n) = A000217(A000290(n)).