A353535 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..5.

EXAMPLE

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.

PROG

(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)

CROSSREFS

Cf. A352969, A353536.

Sequence in context: A006536 A057297 A005530 * A072191 A118324 A060421

Adjacent sequences: A353532 A353533 A353534 * A353536 A353537 A353538

KEYWORD

nonn,hard,more

AUTHOR

Hugo Pfoertner, Apr 22 2022

STATUS

approved