0,3

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.

S. R. Finch, Lehmer's Constant

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Eric Weisstein's World of Mathematics, Lehmer's Constant

Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion

a(n)=floor(c^(2^n)) for n>0, where c=1.385089248334672909882206535871311526236739234374149506334120193387331772... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 29 2002

(PARI) a(n)=if(n<1, 0, a(n-1)^2+a(n-1)+1)

Cf. A002794, A002795, A002665, A030125, A002065, A063573.

Adjacent sequences: A002062 A002063 A002064 this_sequence A002066 A002067 A002068

Sequence in context: A114317 A081299 A117808 this_sequence A087601 A145503 A112093

easy,nice,nonn

N. J. A. Sloane (njas(AT)research.att.com).