A057156 Number of functions from {0,1}^n to {0,1}^n.

1, 4, 256, 16777216, 18446744073709551616, 1461501637330902918203684832716283019655932542976, 39402006196394479212279040100143613805079739270465446667948293404245721771497210611414266254884915640806627990306816

Offset: 0

Author

Henry Bottomley, Aug 15 2000

Keywords

• easy
• nice
• nonn

Comments

a(n) is the number of subdivisions of the Brownian motion on the unit interval at the n-th stage of subdivision. - Stephen Crowley, Apr 12 2007

Examples

			a(1)=4 since the possibilities are f(0)=0, f(1)=0; f(0)=0, f(1)=1; f(0)=1, f(1)=0; f(0)=1, f(1)=1.
For n=3: we need to count maps from a set with 8 points to a set with 8 points.  There are 8^8 such functions, that is, a(3) = 8^8 = 2^24 = 16777216. - _N. J. A. Sloane_, Mar 05 2023
		

References

• François Robert, Discrete Iterations: A Metric Study, Springer-Verlag, 1986, p. 167.
• Norbert Wiener, Nonlinear Problems in Random Theory, MIT Press Classic, 1958, Lecture 1.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..8

Crossrefs

Cf. A000079, A000312, A000722, A001146, A036289, A043322, A057157, A092258, A134880.

Programs

Mathematica

lst={};Do[AppendTo[lst,(2^n)^(2^n)],{n,0,8}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 02 2009 *)

PARI

a(n)=1<<(n<Charles R Greathouse IV, Jan 19 2012

Formula

a(n) = (2^n)^(2^n) = A000312(A000079(n)) = A000079(A036289(n)) = A001146(n)^n = A000722(n) + A057157(n).

Sum_{n>=1} 1/a(n) = A134880. - Amiram Eldar, Nov 15 2020

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 02 2009