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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}.

%I A341736 #21 Feb 26 2022 10:48:20
%S A341736 0,1,7,10,37,40,45,121,124,129,136,301,304,309,316,325,631,634,639,
%T A341736 646,655,666,1177,1180,1185,1192,1201,1212,1225,2017,2020,2025,2032,
%U A341736 2041,2052,2065,2080,3241,3244,3249,3256,3265,3276,3289,3304,3321,4951,4954,4959
%N 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}.
%C A341736 The product in S is computed componentwise.
%C A341736 For the labeling of the elements in S and further information see A341317.
%H A341736 Alois P. Heinz, <a href="/A341736/b341736.txt">Table of n, a(n) for n = 0..10000</a>
%F A341736 a(n) = A341317(n,n) = A341318(n,n).
%F A341736 a(A000217(n)) = A037270(n) = A000217(A000290(n)).
%p A341736 t:= n-> n*(n-1)/2:
%p A341736 f:= n-> ceil((sqrt(1+8*n)-1)/2):
%p A341736 g:= n-> (x-> [x, n-t(x)])(f(n)):
%p A341736 a:= n-> (h-> t(h[1]^2)+h[2]^2)(g(n)):
%p A341736 seq(a(n), n=0..60);
%t A341736 t[n_] := n*(n - 1)/2;
%t A341736 f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
%t A341736 g[n_] := Function[x, {x, n - t[x]}][f[n]];
%t A341736 a[n_] := Function[h, t[h[[1]]^2] + h[[2]]^2][g[n]];
%t A341736 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 26 2022, after _Alois P. Heinz_ *)
%Y A341736 Main diagonal of A341317 and of A341318.
%Y A341736 Cf. A000217, A000290, A037270, A123578, A161680.
%K A341736 nonn
%O A341736 0,3
%A A341736 _Alois P. Heinz_, Feb 17 2021