OFFSET
0,1
COMMENTS
ackb(x,z) =
{ x+2 for z=0
{
{ z for x=0, z>0
{
{ ackb(ackb(x-1,z), z-1) for x,z > 0
This version of the Ackermann function was created with the goal of creating the fastest growth with the least total number of operators, recursive calls, and conditional tests. Check the link for more details.
The reason we take ackb(n,3) is that it is the only sequence that can have its own entry in the OEIS.
a(3) has 7575669 decimal digits and is too big to be included in the data section.
LINKS
Robert Munafo, Versions of Ackermann's functions
FORMULA
a(n) = ackb(x,3) = (3/2) ie3(sqrt(8), x, 8/3) - 1 where ie3(a, b, c) = a^(a^( ... a^c))) (with b copies of a).
For proof, check the link above.
EXAMPLE
a(1) = ackb(1,3) = (3/2) ie3(sqrt(8), 1, 8/3) - 1 = (3/2)sqrt(8)^(8/3) - 1 = (3/2)2^((3/2)(8/3)) - 1 = (3/2)16 - 1 = 23.
MATHEMATICA
ie3[a_, b_, c_] := Nest[a^# &, c, b]; Table[(3/2) ie3[Sqrt@8, x, 8/3] - 1, {x, 0, 2}] (* Michael De Vlieger, Dec 01 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Natan Arie Consigli, Nov 28 2015
STATUS
approved