OFFSET
0,3
COMMENTS
Number of binary trees of height n where for each node the left subtree is at least as high as the right subtree. - Franklin T. Adams-Watters, Feb 08 2007
The next term (a(10)) has 129 digits. - Harvey P. Dale, Jan 24 2016
Number of plane trees where the root has exactly n children and the i-th child of any node has at most i-1 children. - David Eppstein, Dec 18 2021
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
John Cerkan, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, alternative link.
Daniel Duverney, Takeshi Kurosawa, Iekata Shiokawa, Transformation formulas of finite sums into continued fractions, arXiv:1912.12565 [math.NT], 2019.
FORMULA
a(n) ~ c^(2^n), where c = 1.3352454783981919948826893254756974184778316104856161827213437094446034867599... . - Vaclav Kotesovec, May 21 2015
MATHEMATICA
a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1]^2*(1 + 1/a[n - 2]); Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Jul 02 2013 *)
nxt[{t_, a_}]:={t+t*a, t*a}; Transpose[NestList[nxt, {1, 1}, 10]][[2]] (* Harvey P. Dale, Jan 24 2016 *)
PROG
(PARI) a(n)=if(n<2, n >= 0, a(n-1)^2*(1+1/a(n-2)))
(Haskell)
a001697 n = a001697_list !! n
a001697_list = 1 : 1 : f [1, 1] where
f xs@(x:_) = y : f (y : xs) where y = x * sum xs
-- Reinhard Zumkeller, Apr 29 2013
(Magma) [n le 2 select 1 else Self(n-1)^2*(1+1/Self(n-2)): n in [1..12]]; // Vincenzo Librandi, Nov 25 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Additional comments from Michael Somos, May 19 2000
STATUS
approved