A182173 Number of inequivalent expressions involving n operands.

2, 10, 94, 1466, 31814, 887650, 30259198, 1218864842, 56644903958, 2983300619410, 175598066553166, 11423394497044154, 813897286250604326, 63030237104398839490, 5271647928235911880222, 473558482553909252128298, 45473767604938843870986422, 4648336478135316689480390770

Offset

1,1

Comments

Each operand must be used exactly once, and the only allowed operations are addition, subtraction, multiplication, division, and unary minus. Parentheses are permitted. This sequence differs from A140606 by allowing unary minus.

Links

Jingzhe Tang, Table of n, a(n) for n = 1..300 (first 90 terms from David Radcliffe)

David Radcliffe, Python script

Ruud H.G. van Tol, Corrected Python script

Wikipedia, 24 Game

Zhujun Zhang, A Chinese web page on exponential generating function

Zhujun Zhang, A Chinese web page on approximation

Formula

From Zhujun Zhang, Aug 11 2018: (Start)

E.g.f: A(x) = B(x) + C(x) - 2*x, where B(x) = 2*x + exp(C(x)) - 1 - C(x) and C(x) = 2*x + 2*exp(B(x)) - 2*exp(B(x)/2) - B(x).

a(n) ~ (n/(e*b))^n * sqrt(b)*c/n where b=0.16142418303980816579438744831086877555003744810690... and c=1.8772213095052105788245813534431275116981368728916.... (End)

Example

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.

Prog

(PARI) {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

Crossrefs

Cf. A140606.

Sequence in context: A100622 A355782 A103436 * A362643 A160940 A346649

Adjacent sequences: A182170 A182171 A182172 * A182174 A182175 A182176

Keyword

nonn

Author

David Radcliffe, Apr 16 2012

Status

approved