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%I #11 Apr 26 2022 07:22:57
%S 1,2,6,34,458,41846,169022181
%N a(n) is the cardinality of the set S(n) obtained by the following process: Start with the set S(0) = {i}, where i is the imaginary unit. In step n, the set S(n) is the union of all Gaussian integers obtained by the m*(m+1)/2 sums and the m*(m+1)/2 products formed with the pairs of numbers in the Cartesian product S(n-1) x S(n-1) with m = card(S(n-1)).
%H Hugo Pfoertner, <a href="/A353536/a353536.png">Illustration of terms a(0)-a(4)</a>.
%H Hugo Pfoertner, <a href="/A353536/a353536.pdf">Illustration of a(5) = 41846</a>.
%e S(0) = {i}, a(0) = 1;
%e S(1) = {-1, 2*i}, a(1) = 2;
%e S(2) = {-4, -2, 1, -1+2*i, -2*i, 4*i}, a(2) = 6;
%e S(3) = {-16, -8, -6, -4, -3, -2, -1, 1, 2, 4, 8, 16, -8-4*i, -5+2*i, -4-2*i, -4+4*i, -3-4*i, -3+2*i, -2-2*i, -2+4*i, -1+2*i, -1+6*i, -16*i, -8*i, -4*i, -2*i, 2*i, 4*i, 8*i, 1-2*i, 1+4*i, 2-4*i, 4-8*i, 4+2*i}, a(3) = 34.
%o (PARI) a353536(nmax) = {my(v=[I],m=#v); print1(m,", "); for(n=1,nmax, my(L=m*(m+1), w=vector(L), k=0); for(i=1,#v, for(j=i,#v, w[k++]=v[i]+v[j]; w[k++]=v[i]*v[j])); v=Set(w); m=#v; print1(m,", "))};
%o a353536(5)
%Y Cf. A352969, A353535.
%K nonn,hard,more
%O 0,2
%A _Hugo Pfoertner_, Apr 26 2022
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