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A003095
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a(n) = a(n-1)^2 + 1.
(Formerly M1544)
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23
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0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 (jvospost3(AT)gmail.com), Jul 21 2005
Apart from the initial term a subsequence of A008318. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 17 2008
Partial sums of A001699. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 17 2010]
Corresponds to the second and second last diagonals of A119687. [From John M. Campbell, Jul 25 2011]
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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.
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LINKS
| 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.
C. Lenormand, Arbres et permutations II, see p. 6
Index entries for sequences of form a(n+1)=a(n)^2 + ...
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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... - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (gerald.mcgarvey(AT)comcast.net), Nov 17 2007
a(n) = SUM[i=1..n] A001699(i). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 17 2010]
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MATHEMATICA
| NestList[#^2+1&, 0, 10] [From Harvey P. Dale, Dec. 17, 2010]
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CROSSREFS
| Cf. A038044.
Cf. A001699, A056207, A004019.
A143848, A143849. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 03 2008]
Cf. A137560, which enumerates binary trees of height less than n and exactly j leaf nodes. [From Robert Munafo (mrob27(AT)gmail.com), Nov 03 2009]
Sequence in context: A111195 A167007 A064006 * A023362 A138613 A090744
Adjacent sequences: A003092 A003093 A003094 * A003096 A003097 A003098
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. P. Stanley
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EXTENSIONS
| Additional comments from Cyril Banderier (Cyril.Banderier(AT)inria.fr), Jun 05 2000
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