A002065 a(n+1) = a(n)^2 + a(n) + 1. (Formerly M2961 N1197)

0, 1, 3, 13, 183, 33673, 1133904603, 1285739649838492213, 1653126447166808570252515315100129583, 2732827050322355127169206170438813672515557678636778921646668538491883473

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, alternative link.

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

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

Formula

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

a(n) = (A232806(n) - 1)/2 = (A232806(n-1)^2 + 3)/4. - Peter Bala, Mar 28 2018

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

Cf. A002665, A002794, A002795, A030125, A063573, A232806.

Sequence in context: A368499 A081299 A117808 * A302143 A087601 A145503

Adjacent sequences: A002062 A002063 A002064 * A002066 A002067 A002068

Keyword

easy,nice,nonn

Author

N. J. A. Sloane

Status

approved