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A382928 Start with {1, x}, then at each step replace it with the set of all pairwise products and sums of its elements (an element can be paired with itself). a(n) gives the number of elements after n-th step.

%I A382928 #28 Apr 11 2025 16:19:54
%S A382928 2,6,28,436,90385,4017112742
%N A382928 Start with {1, x}, then at each step replace it with the set of all pairwise products and sums of its elements (an element can be paired with itself). a(n) gives the number of elements after n-th step.
%C A382928 a(n) is the cardinality of s(n) where s(0) = {1, x} and s(n) contains a+b, a*b and a for every a and b in s(n-1).
%C A382928 Related to A352969 which has {1} as the starting set as opposed to {1, x}.
%e A382928 a(0) = 2 corresponding to 1 and x.
%e A382928 a(1) = 6 since from 1 and x, we can reach 4 other polynomials namely: x^2, 2x, x+1 and 2.
%t A382928 s[0]={1,x}; s[n_]:=s[n]= Union[Expand/@ Flatten@ ({Plus@@#, Times@@#}& /@ Tuples[s[n-1],2])]; a[n_]:=Length@ s@ n; a/@ Range[0,4] (* _Giovanni Resta_, Apr 10 2025 *)
%o A382928 (PARI) lista(nn) = {my(v = [1, x]); print1(#v, ", "); for (n=1, nn, v = setunion(setbinop((x, y)->(x+y), v), setbinop((x, y)->(x*y), v)); print1(#v, ", "); ); } \\ _Yifan Xie_, Apr 09 2025
%Y A382928 Cf. A352969.
%K A382928 nonn,more,hard
%O A382928 0,1
%A A382928 _Bryle Morga_, Apr 09 2025
%E A382928 a(5) from _Bert Dobbelaere_, Apr 11 2025