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EXAMPLE
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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.
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