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)).
1, 2, 6, 38, 1078, 749674, 380157661301
Offset: 0
Examples
S(0) = {x}, a(0) = 1;
S(1) = {2*x, x^2}, a(1) = 2;
S(2) = {4*x, 2*x^2, 4*x^2, x^2 + 2*x, 2*x^3, x^4}, a(2) = 6;
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.
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.
Programs
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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 lista(5)
Extensions
a(6) from Bert Dobbelaere, May 22 2025