This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353535 #15 May 23 2025 12:12:05
%S A353535 1,2,6,38,1078,749674,380157661301
%N A353535 a(n) is the cardinality of the set S(n) obtained by the following process: Start with the set S(0) of a single monomial {x}. In step n, the set S(n) is the union of all polynomials obtained by the m*(m+1)/2 sums and the m*(m+1)/2 products formed with the pairs of polynomials in the Cartesian product S(n-1) x S(n-1) with m = card(S(n-1)).
%e A353535 S(0) = {x}, a(0) = 1;
%e A353535 S(1) = {2*x, x^2}, a(1) = 2;
%e A353535 S(2) = {4*x, 2*x^2, 4*x^2, x^2 + 2*x, 2*x^3, x^4}, a(2) = 6;
%e A353535 S(3) = {8*x, 4*x^2, 6*x^2, 8*x^2, 16*x^2, 3*x^2 + 2*x, 5*x^2 + 2*x, 2*x^2 + 4*x, 4*x^2 + 4*x, x^2 + 6*x, 4*x^3, 8*x^3, 16*x^3, 2*x^3 + 2*x^2, 2*x^3 + 4*x^2, 4*x^3 + 8*x^2, 2*x^3 + x^2 + 2*x, 2*x^3 + 4*x, 2*x^4, 4*x^4, 8*x^4, 16*x^4, x^4 + 2*x^3, 2*x^4 + 4*x^3, 4*x^4 + 8*x^3, x^4 + 2*x^2, x^4 + 4*x^2, x^4 + 4*x^3 + 4*x^2, x^4 + x^2 + 2*x, x^4 + 4*x, 4*x^5, 8*x^5, 2*x^5 + 4*x^4, 2*x^6, 4*x^6, x^6 + 2*x^5, 2*x^7, x^8}, a(3) = 38, with 3 collapsing products 4*x^5, 4*x^6, 8*x^4 and 1 collapsing sum 2*x^2 + 4*x. m=6, 2*(7*6/2) - 3 - 1 = 38.
%e A353535 a(4) = 1078 is formed by 611 distinct polynomials from the sums and 487 distinct polynomials from the products. There are 20 polynomials in the intersection of the sum set and the product set: {32*x^3, 16*x^4, 24*x^4, 32*x^4, 12*x^4 + 8*x^3, 20*x^4 + 8*x^3, 8*x^4 + 16*x^3, 16*x^4 + 16*x^3, 4*x^4 + 24*x^3, 16*x^5, 8*x^5 + 8*x^4, 8*x^5 + 16*x^4, 8*x^5 + 4*x^4 + 8*x^3, 8*x^5 + 16*x^3, 8*x^6, 4*x^6 + 8*x^5, 4*x^6 + 8*x^4, 4*x^6 + 16*x^4, 4*x^6 + 4*x^4 + 8*x^3, 4*x^6 + 16*x^3}, thus 611 + 487 - 20 = 1078.
%o A353535 (PARI) lista(nn) = {my(v = ['z]); print1(#v, ", "); for (n=1, nn, v = setunion(setbinop((x, y)->(x+y), v), setbinop((x, y)->(x*y), v)); print1(#v, ", "); ); } \\ after _Michel Marcus_ in A352969
%o A353535 lista(5)
%Y A353535 Cf. A352969, A353536.
%K A353535 nonn,hard,more
%O A353535 0,2
%A A353535 _Hugo Pfoertner_, Apr 22 2022
%E A353535 a(6) from _Bert Dobbelaere_, May 22 2025