%I #44 Nov 23 2023 15:59:12

%S 2,12,228,62052,4180832868,18201642257939067492,

%T 338021701687178649306251838479209230948,

%U 115407456979036362321626052309736660160730393295399201179594209600531491615332

%N a(n) is the integer corresponding to a bit-string representation of the von Neumann ordinal representation of n, with largest sets listed first, and with '{' represented by the bit 1, '}' represented by the bit zero, and ignoring commas.

%C Since the von Neumann ordinals begin with 0={}, it seems appropriate to have an offset of 0.

%C The sequence grows super-exponentially.

%C Similar to A092124, except that A092124 reverses the order of the elements in the ordinal.

%C 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

%H Kit Scriven, <a href="/A333447/b333447.txt">Table of n, a(n) for n = 0..10</a>

%H Johann von Neumann, <a href="http://acta.bibl.u-szeged.hu/13294/1/math_001_199-208.pdf">Zur Einführung der transfiniten Zahlen</a>, Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, 1 (1923), 199-208.

%F a(0) = 2, a(n) = 2^(2^(n+1)-1) - 2^(2^(n)-1) + a(n-1)*(2^(2^(n)-1) + 1).

%e A table demonstrating the von Neumann ordinals of the first three integers, their corresponding bit strings, and their sequence values is as follows:

%e n set notation bit string a(n)

%e 0 {} 10 2

%e 1 {{}} 1100 12

%e 2 {{{}}{}} 11100100 228

%e 3 {{{{}}{}}{{}}{}} 1111001001100100 62052

%t With[{nmax=8},Map[FromDigits[#,2]&,NestList["1"<>#<>StringTake[#,{2,-2}]<>"0"&,"10",nmax]]] (* _Paolo Xausa_, Nov 21 2023 *)

%o (Python)

%o def fBinDigit(n):

%o return 2**(2**(n+1) - 1)

%o def a333447(n):

%o if n==0:

%o return 2

%o else:

%o prevAsEle = a333447(n-1) * fBinDigit(n-1)

%o restOfEle = a333447(n-1) - fBinDigit(n-1)

%o return fBinDigit(n)+prevAsEle+restOfEle

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o 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

%Y Cf. A079559, A092124, A308187.

%K nonn,base

%O 0,1

%A _Kit Scriven_, Mar 21 2020