A182173
Number of inequivalent expressions involving n operands.
Original entry on oeis.org
2, 10, 94, 1466, 31814, 887650, 30259198, 1218864842, 56644903958, 2983300619410, 175598066553166, 11423394497044154, 813897286250604326, 63030237104398839490, 5271647928235911880222, 473558482553909252128298, 45473767604938843870986422, 4648336478135316689480390770
Offset: 1
When n=2, there are 10 inequivalent expressions: a+b, a-b, b-a, -a-b, a*b, -a*b, a/b, -a/b, b/a, -b/a.
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{a(n) = my(A,B=x +x*O(x^n),C=x +x*O(x^n)); for(i=1,n, B = 2*x + exp(C) - 1 - C; C = 2*x + 2*exp(B) - 2*exp(B/2) - B ); A = B + C - 2*x; n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", ")) \\ Paul D. Hanna, Aug 12 2018 - After formula by Zhujun Zhang
A131655
Maximal number of different numbers that can be formed from n numbers using addition, subtraction, multiplication and division, when each number is used no more than once in a given expression.
Original entry on oeis.org
1, 8, 89, 1482, 33737, 974860, 34078585, 1397335710, 65753020657, 3493075453208, 206827445591945, 13508998509463890, 965035059918868889, 74859785094041370724, 6267261735949374860857, 563289843479215734940710, 54100963704277853262416833
Offset: 1
Daniel Rainjonneau (d.rainjonneau(AT)wanadoo.fr), Sep 11 2007
a(2) = 8 : from two variables x and y we obtain the eight functions x, y, x+y, x*y, x-y, y-x, x/y and y/x.
A247982
Number of inequivalent expressions involving n operands, ignoring sign.
Original entry on oeis.org
1, 5, 47, 733, 15907, 443825, 15129599, 609432421, 28322451979, 1491650309705, 87799033276583, 5711697248522077, 406948643125302163, 31515118552199419745, 2635823964117955940111, 236779241276954626064149, 22736883802469421935493211
Offset: 1
A351922
Consider well-formed strings consisting of n operands, n - 1 binary operators +, -, *, /, ^, and n-1 pairs of parentheses, and call two such strings equivalent if they are algebraically identical; a(n) is the number of equivalence classes.
Original entry on oeis.org
1, 8, 146, 4294
Offset: 1
For n = 2 the a(2) = 8 expressions are a+b, a-b, b-a, a*b, a/b, b/a, a^b, b^a. b+a does not count because it is equivalent to a+b.
For n=3, the a(3)=146 expressions (as rendered canonically by fullratsimp/fortran in maxima) are: a*b-a*c; a*b-b*c; (a*b)/c; (a-b)*c; (a-b)^c; a*b*c; a*b-c; a*b^c; a-b*c; a-b^c; a/(b*c); a/b^c; a^(b*c); a^(b-c); a^(b/c); a^b*c; a^b-c; a^b/c; a^b^c; a^b*c^b; a^b/c^b; a*c+a*b; a*c-a*b; (a*c)/b; (a-c)^b; -a/(c-b); a*c+b; a*c-b; a*c^b; a-c^b; a/(c+b); a/(c-b); a/c^b; a^(c+b); a^(c-b); a^(c/b); a^c*b; a^c-b; a^c/b; a^c^b; (a*c+b)/a; (a*c-b)/a; -(a*c-b)/a; (a*c+b)/c; (a*c-b)/c; -(a*c-b)/c; a^c*b^c; a^c/b^c; (b+a)*c; (b+a)/c; (b+a)^c; (b-a)*c; (b-a)/c; (b-a)^c; -(b-a)/c; b+a^c; b-a*c; b-a^c; b/(a*c); b/a^c; b^(a*c); b^(a-c); b^(a/c); b^a*c; b^a-c; b^a/c; b^a^c; b^a*c^a; b^a/c^a; (b*c)/a; (b-c)^a; -b/(c-a); b*c+a; b*c-a; b*c^a; b-c^a; b/(c+a); b/(c-a); b/c^a; b^(c+a); b^(c-a); b^(c/a); b^c+a; b^c-a; b^c/a; b^c^a; (b*c+a)/b; (b*c-a)/b; -(b*c-a)/b; b*c+a*b; b*c-a*b; (b*c+a)/c; (b*c-a)/c; -(b*c-a)/c; b^c/a^c; (c+a)/b; (c+a)^b; (c-a)/b; (c-a)^b; -(c-a)/b; c+a*b; c+a^b; c-a*b; c-a^b; c/(a*b); c/a^b; c^(a*b); c^(a-b); c^(a/b); c^a+b; c^a-b; c^a/b; c^a^b; (c+a*b)/a; (c-a*b)/a; -(c-a*b)/a; c^a/b^a; (c+a*b)/b; (c-a*b)/b; -(c-a*b)/b; (-c)+b+a; (-c)+b-a; (-c)-b+a; (c+b)/a; (c+b)^a; (c-b)/a; (c-b)^a; -(c-b)/a; -c/(b-a); c+b+a; c+b-a; c+b^a; c-b+a; c-b-a; c-b^a; c/(b+a); c/(b-a); c/b^a; c^(b+a); c^(b-a); c^(b/a); c^b+a; c^b-a; c^b/a; c^b^a; c^b/a^b.
Cf.
A140606 (which does not allow ^).
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