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A092124 a(0) = 2, a(n) = (2^(2^n)+2)*a(n-1) for n>0.

%I A092124 #19 Aug 04 2025 15:05:58
%S A092124 2,12,216,55728,3652301664,15686516209310983872,
%T A092124 289365149921256212111714425927549504896,
%U A092124 98465858119637274097902770931519409290135390781788892125023848289699334298368
%N A092124 a(0) = 2, a(n) = (2^(2^n)+2)*a(n-1) for n>0.
%C A092124 In binary representation a(n) can be interpreted as an expression to represent n according to John von Neumann's definition of natural numbers: braces are coded as 1 and 0 and the empty set as 10={};
%C A092124 a(n) = (A001146(n)+2)*a(n-1) = 2*(A058891(n)+1)*a(n-1).
%H A092124 Hugo Pfoertner, <a href="/A092124/b092124.txt">Table of n, a(n) for n = 0..10</a>
%e A092124 a(3)=55728='1101100110110000' -> {{}{{}}{{}{{}}}} -> {{},{{}},{{},{{}}}} -> {0,{0},{0,{0}}} -> {0,1,{0,1}} -> {0,1,2} -> A001477(3)=3.
%t A092124 RecurrenceTable[{a[0]==2,a[n]==(2^(2^n)+2)a[n-1]},a,{n,8}] (* _Harvey P. Dale_, Nov 15 2020 *)
%t A092124 nxt[{n_,a_}]:={n+1,(2^2^(n+1)+2)a}; NestList[nxt,{0,2},8][[;;,2]] (* _Harvey P. Dale_, Aug 11 2023 *)
%o A092124 (Python)
%o A092124 from functools import lru_cache
%o A092124 @lru_cache(maxsize=None)
%o A092124 def A092124(n): return ((1<<(1<<n))+2)*A092124(n-1) if n else 2 # _Chai Wah Wu_, Nov 23 2023
%Y A092124 Cf. A001146, A058891, A333447, A367033.
%K A092124 nonn
%O A092124 0,1
%A A092124 _Reinhard Zumkeller_, Mar 30 2004