%I A007018 M1713
%S A007018 1,2,6,42,1806,3263442,10650056950806,113423713055421844361000442,
%T A007018 12864938683278671740537145998360961546653259485195806
%N A007018 a(n)=a(n-1)^2+a(n-1), a(0)=1.
%C A007018 Number of ordered trees having nodes of outdegree 0,1,2 and such that 
               all leaves are at level n. Example: a(2)=6 because, denoting by I 
               a path of length 2 and by Y a Y-shaped tree with 3 edges, we have 
               I, Y, I*I, I*Y, Y*I, Y*Y, where * denotes identification of the roots. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2002
%C A007018 a(n) has at least n different prime factors [Saidak]
%C A007018 Subsequence of square-free numbers (A005117). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 15 2004
%C A007018 For prime factors see A007996.
%C A007018 Curtiss shows that if the reciprocal sum of the multiset S = {x_1, x_2, 
               ..., x_n} is 1, then max(S) <= a(n). - Charles R Greathouse IV, Feb 
               28 2007
%C A007018 The number of reduced ZBDDs for Boolean functions of n variables in which 
               there is no zero sink. (ZBDDs are "zero-suppressed binary decision 
               diagrams.") For example, a(2)=6 because of the 2-variable functions 
               whose truth tables are 1000, 1010, 1011, 1100, 1110, 1111. - D. E. 
               Knuth, Jun 04 2007
%C A007018 Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, 
               it is easy to show that a(n) is the integer just a tiny bit below 
               the real number theta^{2^n}-1/2, where theta =~ 1.597910218 is the 
               exponential of the rapidly convergent series ln(3/2)+sum_{n >= 0} 
               ln(1+(2a_n+1)^{-2}). For example, theta^64-1/2 =~ 3263442.0000000383. 
               - D. E. Knuth, Jun 04 2007
%C A007018 a(n) = A139145(2^(n+1) - 1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 10 2008
%C A007018 a(n+1) = a(n) th oblong (or promic, pronic, or heteromecic) numbers (A002378). 
               a(n+1) = A002378(a(n)) = A002378(a(n-1)) * (A002378(a(n-1)) + 1). 
               [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 13 2009]
%D A007018 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007018 D. R. Curtiss, "On Kellogg's Diophantine Problem". The American Mathematical 
               Monthly Vol. 29, No. 10 (1922), pp. 380-387.
%D A007018 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 94.
%D A007018 M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM 
               Review, SIAM, 1990; see p. 577.
%D A007018 Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 
               2006
%D A007018 A. Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, 
               1 (1995) pp. 425-, esp. p. 434.
%H A007018 N. J. A. Sloane, <a href="b007018.txt">Table of n, a(n) for n = 0..12</
               a>
%H A007018 <a href="Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 
               + ...</a>
%F A007018 a(n) = A000058(n)-1 = A000058(n-1)^2-A000058(n-1) = 1/(1-sum{j<n}[1/A000058(j)]) 
               where A000058 is Sylvester's sequence. - Henry Bottomley (se16(AT)btinternet.com), 
               Jul 23 2001
%F A007018 a(n) = floor(c^(2^n)) where c=1.597910218031873178338070118157... - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002
%F A007018 a(1)=1, a(n) = prod(k=1, n-1, a(k)+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Sep 13 2003
%F A007018 If an (additional) initial 1 is inserted, a(n) = sum_{k<n} a(k)^2. [From 
               Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 11 2009]
%t A007018 a=1;lst={};Do[a=a^2+a;AppendTo[lst,a],{n,0,9}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Oct 20 2009]
%o A007018 (PARI) a(n)=if(n<1,n>=0,a(n-1)+a(n-1)^2)
%Y A007018 Cf. A003687.
%Y A007018 Cf. A011782. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 11 2009]
%Y A007018 Sequence in context: A123137 A014117 A054377 this_sequence A100016 A000610 
               A023363
%Y A007018 Adjacent sequences: A007015 A007016 A007017 this_sequence A007019 A007020 
               A007021
%K A007018 nonn,nice,easy
%O A007018 0,2
%A A007018 N. J. A. Sloane (njas(AT)research.att.com).
%E A007018 The next term is too large to include.