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%I #52 Oct 26 2021 16:20:47
%S 0,1,3,11,2059
%N a(0) = 0; a(n+1) = a(n) + 2^a(n).
%C First impartial game with value n, using natural enumeration of impartial games.
%C The natural 1-1 correspondence between nonnegative numbers and hereditarily finite sets is given by f(A)=sum over members m of A of 2^f(m). A set can be considered an impartial game where the legal moves are the members. The value of an impartial game is always an ordinal (for finite games, an integer).
%C The next term, a(5) = 2^2059 + 2059, has 620 decimal digits and is too large to include. - _Olivier Gérard_, Jun 26 2001
%C Positions of records in A103318. - _N. J. A. Sloane_ and _David Applegate_, Mar 21 2005
%C The first n terms in this sequence form the lexicographically earliest n-vertex clique in the Ackermann-Rado encoding of the Rado graph (an infinite graph in which vertex i is adjacent to vertex j, with i<j, when the i-th bit of the binary representation of j is nonzero). - _David Eppstein_, Aug 22 2014
%C This sequence was used by Spiro to bound the density of refactorable numbers (A033950). - _David Eppstein_, Aug 22 2014
%D J. H. Conway, On Numbers and Games, Academic Press.
%H N. J. A. Sloane, <a href="/A034797/b034797.txt">Table of n, a(n) for n = 0..5</a>
%H Wilhelm Ackermann, <a href="http://dx.doi.org/10.1007/BF01594179">Die Widerspruchsfreiheit der allgemeinen Mengenlehre</a>, Math. Ann. 114 (1939), no. 1, 305-315.
%H David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Sloane/sloane300.html">Sloping binary numbers: a new sequence related to the binary numbers</a>, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [<a href="http://neilsloane.com/doc/slopey.pdf">pdf</a>, <a href="http://neilsloane.com/doc/slopey.ps">ps</a>].
%H O. Kurganskyy and I. Potapov, <a href="http://arxiv.org/abs/1510.04121">Reachability problems for PAMs</a>, arXiv preprint arXiv:1510.04121 [cs.NA], 2015.
%H Richard Rado, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa9/aa9133.pdf">Universal graphs and universal functions</a>, Acta Arith. 9 (1964), 331-340.
%H Claudia Spiro, <a href="http://dx.doi.org/10.1016/0022-314X(85)90012-5">How often is the number of divisors of n a divisor of n?</a>, J. Number Theory 21 (1985), no. 1, 81-100.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rado_graph">Rado graph</a>.
%t a=0;lst={a};Do[AppendTo[lst,a+=2^a],{n,0,4}];lst (* _Vladimir Joseph Stephan Orlovsky_, May 06 2010 *)
%t NestList[#+2^#&,0,5] (* _Harvey P. Dale_, Mar 22 2020 *)
%Y Cf. A034798, A103318.
%K nonn
%O 0,3
%A Joseph Shipman (shipman(AT)savera.com)
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