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.

2, 6, 28, 436, 90385, 4017112742

Offset: 0

Author

Bryle Morga, Apr 09 2025

Keywords

• nonn
• more
• hard

Comments

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).

Related to A352969 which has {1} as the starting set as opposed to {1, x}.

Examples

			a(0) = 2 corresponding to 1 and x.
a(1) = 6 since from 1 and x, we can reach 4 other polynomials namely: x^2, 2x, x+1 and 2.
		

Crossrefs

Cf. A352969.

Programs

Mathematica

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 *)

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

Extensions

a(5) from Bert Dobbelaere, Apr 11 2025