OFFSET
0,3
COMMENTS
a(n) is the 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..., gives the number h(n) of these trees whose 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) gives sequence 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
This is a divisibility sequence, that is, if n divides m, then a(n) divides a(m). This is a particular case of the result: if p(x) is an integral polynomial then the sequence of n-th iterates p^n(x) (:= p(p^(n-1)(x)) with p^1(x) := p(x)), n = 1,2,..., of p(x) evaluated at x = 0 is a divisibility sequence. In this case p(x) = 1 + x + x^2. - Peter Bala, Mar 28 2018
REFERENCES
Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
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.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Steven R. Finch, Lehmer's Constant [Broken link]
Steven R. Finch, Lehmer's Constant [From the Wayback machine]
Stan C. Kalman and Barry L. Kwasny, Tail-recursive distributed representations and simple recurrent networks, Connection Science, 7 (1995), 61-80.
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971
Eric Weisstein's World of Mathematics, Lehmer's Constant
Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion
Wikipedia, Herbrand Structure
J. W. Wrench, Jr., Letters to N. J. A. Sloane, Feb 1974
FORMULA
a(n) = floor(c^(2^n)) for n > 0, where c = 1.385089248334672909882206535871311526236739234374149506334120193387331772... - Benoit Cloitre, Nov 29 2002
MATHEMATICA
f[x_] := 1 + x + x^2 ; NestList[f, 1, 7] (* Geoffrey Critzer, May 04 2010 *)
PROG
(PARI) a(n)=if(n<1, 0, a(n-1)^2+a(n-1)+1)
(Magma) [n le 1 select 0 else Self(n-1)^2 + Self(n-1) + 1: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015
(Maxima) a(n) := if n = 0 then 1 else a(n-1)^2+a(n-1)+1 $
makelist(a(n), n, 0, 8); /* Emanuele Munarini, Mar 23 2017 */
CROSSREFS
KEYWORD
easy,nice,nonn,changed
AUTHOR
STATUS
approved