%I A002065 M2961 N1197
%S A002065 0,1,3,13,183,33673,1133904603,1285739649838492213,
%T A002065 1653126447166808570252515315100129583,
%U A002065 2732827050322355127169206170438813672515557678636778921646668538491883473
%N A002065 a(n+1) = a(n)^2 + a(n) + 1.
%C A002065 a(n) = number of trees of height <= n, generated by unary and binary 
               composition: S = x + (S) + (S,S) = x + (x) + (x,x) + (x,(x)) + ((x),
               x) + ((x)) + ((x),(x)) + (x,(x,x)) + ((x,x),x) + ((x),(x,x)) + ((x,
               x),(x)) + ((x,x)) + ((x,x),(x,x)) + ... (x is of height 1); the first 
               difference sequence (beginning with 1), 1 2 10 170 33490 1133870930..., 
               give the number h(n) of these trees whose the height is n, h(n + 
               1) = h(n) + h(n)*h(n) + 2h(n)*a(n-1), h(1) = 1; As h(n + 1)/h(n) 
               = 1 + a(n) + a(n-1), 1 2 10 = 2*5 170 = 2*5*17 33490 = 2*5*17*197 
               1133870930 = 2*5*17*197*33877... - Claude Lenormand (claude.lenormand(AT)free.fr), 
               Sep 05 2001
%D A002065 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002065 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002065 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
%D A002065 D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. 
               J., 4 (1935), 323-340.
%H A002065 A. V. Aho and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 
               11 (1973), 429-437.
%H A002065 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/lehmer/lehmer.html">
               Lehmer's Constant</a>
%H A002065 <a href="Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 
               + ...</a>
%H A002065 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LehmersConstant.html">Lehmer's Constant</a>
%H A002065 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LehmerCotangentExpansion.html">Lehmer Cotangent Expansion</a>
%F A002065 a(n)=floor(c^(2^n)) for n>0, where c=1.3850892483346729098822065358713115262367392343741495063341201933873317\
               72... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 29 2002
%o A002065 (PARI) a(n)=if(n<1,0,a(n-1)^2+a(n-1)+1)
%Y A002065 Cf. A002794, A002795, A002665, A030125, A002065, A063573.
%Y A002065 Sequence in context: A114317 A081299 A117808 this_sequence A087601 A145503 
               A112093
%Y A002065 Adjacent sequences: A002062 A002063 A002064 this_sequence A002066 A002067 
               A002068
%K A002065 easy,nice,nonn
%O A002065 0,3
%A A002065 N. J. A. Sloane (njas(AT)research.att.com).