OFFSET
0,1
COMMENTS
Since the von Neumann ordinals begin with 0={}, it seems appropriate to have an offset of 0.
The sequence grows super-exponentially.
The binary expansion of a(n-1) has length 2^n and consists of n 1's followed by the leading terms of A308187. - Andrey Zabolotskiy, Mar 21 2020
LINKS
Kit Scriven, Table of n, a(n) for n = 0..10
Johann von Neumann, Zur Einführung der transfiniten Zahlen, Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, 1 (1923), 199-208.
FORMULA
a(0) = 2, a(n) = 2^(2^(n+1)-1) - 2^(2^(n)-1) + a(n-1)*(2^(2^(n)-1) + 1).
EXAMPLE
A table demonstrating the von Neumann ordinals of the first three integers, their corresponding bit strings, and their sequence values is as follows:
n set notation bit string a(n)
0 {} 10 2
1 {{}} 1100 12
2 {{{}}{}} 11100100 228
3 {{{{}}{}}{{}}{}} 1111001001100100 62052
MATHEMATICA
With[{nmax=8}, Map[FromDigits[#, 2]&, NestList["1"<>#<>StringTake[#, {2, -2}]<>"0"&, "10", nmax]]] (* Paolo Xausa, Nov 21 2023 *)
PROG
(Python)
def fBinDigit(n):
return 2**(2**(n+1) - 1)
def a333447(n):
if n==0:
return 2
else:
prevAsEle = a333447(n-1) * fBinDigit(n-1)
restOfEle = a333447(n-1) - fBinDigit(n-1)
return fBinDigit(n)+prevAsEle+restOfEle
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A333447(n): return (1<<((m:=1<<n)<<1)-1)-(k:=1<<m-1)+(k+1)*A333447(n-1) if n else 2 # Chai Wah Wu, Nov 23 2023
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Kit Scriven, Mar 21 2020
STATUS
approved