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A003095 a(n) = a(n-1)^2 + 1.
(Formerly M1544)
27
0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of binary trees of height less than n.

The rightmost digits cycle (0,1,2,5,6,7,0,1,2,5,6,7,...). a(n) is prime for n = 2, 3, 5, ... a(n) is semiprime for n = 4, ... - Jonathan Vos Post, Jul 21 2005

Apart from the initial term a subsequence of A008318. - Reinhard Zumkeller, Jan 17 2008

Partial sums of A001699. - Jonathan Vos Post, Feb 17 2010

Corresponds to the second and second last diagonals of A119687. - John M. Campbell, Jul 25 2011

This is a divisibility sequence. - Michael Somos, Jan 01 2013

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.

R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 49-89.

R. Penrose, The Emperor's New Mind, Oxford, 1989, p. 122

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

M. Tainiter, Algebraic approach to stopping variable problems: Representation theory and applications. J. Combinatorial Theory 9 1970 148-161.

LINKS

Table of n, a(n) for n=0..10.

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

P. Flajolet and A. M. Odlyzko, Limit distributions of coefficients of iterates of polynomials with applications to combinatorial enumerations, Math. Proc. Camb. Phil. Soc., 96 (1984), 237-253.

Spencer Hamblen, Rafe Jones, and Kalyani Madhu, The density of primes in orbits of z^d + c, to appear, Int. Math. Res. Not., c. 2015.

C. Lenormand, Arbres et permutations II, see p. 6

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

FORMULA

a_n=B_{n-1}(1) where B_n(x)=1+xB_{n-1}(x)^2 is the generating function of trees of height <= n.

a(n) is asymptotic to c^(2^n) where c=1.2259024435287485386279474959130085213... (see A076949). - Benoit Cloitre, Nov 27 2002

c = b^(1/4) where b is the constant in Bottomley's formula in A004019. a(n) appears very asymptotic to c^(2^n) - Sum(k=1,infinity, A088674[k]/(2*c^(2^n))^(2*k-1)). - Gerald McGarvey, Nov 17 2007

a(n) = Sum_{i=1..n} A001699(i). - Jonathan Vos Post, Feb 17 2010

EXAMPLE

x + 2*x^2 + 5*x^3 + 26*x^4 + 677*x^5 + 458330*x^6 +210066388901*x^7 + ...

MATHEMATICA

NestList[#^2+1&, 0, 10] (* Harvey P. Dale, Dec 17 2010 *)

PROG

{a(n) = if( n<1, 0, 1 + a(n-1)^2)} /* Michael Somos, Jan 01 2013 */

CROSSREFS

Cf. A038044, A001699, A056207, A004019, A143848, A143849.

Cf. A137560, which enumerates binary trees of height less than n and exactly j leaf nodes. [From Robert Munafo, Nov 03 2009]

Cf. A076949.

Cf. A247981, A248218, A248219.

Sequence in context: A111195 A167007 A064006 * A023362 A138613 A090744

Adjacent sequences:  A003092 A003093 A003094 * A003096 A003097 A003098

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane and Richard Stanley

EXTENSIONS

Additional comments from Cyril Banderier, Jun 05 2000

Minor edits by Vaclav Kotesovec, Oct 04 2014

STATUS

approved

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Last modified October 24 08:42 EDT 2014. Contains 248516 sequences.