0,2

Number of ordered trees having nodes of outdegree 0,1,2 and such that all leaves are at level n. Example: a(2)=6 because, denoting by I a path of length 2 and by Y a Y-shaped tree with 3 edges, we have I, Y, I*I, I*Y, Y*I, Y*Y, where * denotes identification of the roots. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2002

a(n) has at least n different prime factors [Saidak]

Subsequence of square-free numbers (A005117). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 15 2004

For prime factors see A007996.

Curtiss shows that if the reciprocal sum of the multiset S = {x_1, x_2, ..., x_n} is 1, then max(S) <= a(n). - Charles R Greathouse IV, Feb 28 2007

The number of reduced ZBDDs for Boolean functions of n variables in which there is no zero sink. (ZBDDs are "zero-suppressed binary decision diagrams.") For example, a(2)=6 because of the 2-variable functions whose truth tables are 1000, 1010, 1011, 1100, 1110, 1111. - D. E. Knuth, Jun 04 2007

Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, it is easy to show that a(n) is the integer just a tiny bit below the real number theta^{2^n}-1/2, where theta =~ 1.597910218 is the exponential of the rapidly convergent series ln(3/2)+sum_{n >= 0} ln(1+(2a_n+1)^{-2}). For example, theta^64-1/2 =~ 3263442.0000000383. - D. E. Knuth, Jun 04 2007

a(n) = A139145(2^(n+1) - 1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 10 2008

a(n+1) = a(n) th oblong (or promic, pronic, or heteromecic) numbers (A002378). a(n+1) = A002378(a(n)) = A002378(a(n-1)) * (A002378(a(n-1)) + 1). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Sep 13 2009]

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

D. R. Curtiss, "On Kellogg's Diophantine Problem". The American Mathematical Monthly Vol. 29, No. 10 (1922), pp. 380-387.

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 94.

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see p. 577.

Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 2006

A. Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, 1 (1995) pp. 425-, esp. p. 434.

N. J. A. Sloane, Table of n, a(n) for n = 0..12

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

a(n) = A000058(n)-1 = A000058(n-1)^2-A000058(n-1) = 1/(1-sum{j<n}[1/A000058(j)]) where A000058 is Sylvester's sequence. - Henry Bottomley (se16(AT)btinternet.com), Jul 23 2001

a(n) = floor(c^(2^n)) where c=1.597910218031873178338070118157... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 06 2002

a(1)=1, a(n) = prod(k=1, n-1, a(k)+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 13 2003

If an (additional) initial 1 is inserted, a(n) = sum_{k<n} a(k)^2. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 11 2009]

a=1; lst={}; Do[a=a^2+a; AppendTo[lst, a], {n, 0, 9}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 20 2009]

(PARI) a(n)=if(n<1, n>=0, a(n-1)+a(n-1)^2)

Cf. A003687.

Cf. A011782. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 11 2009]

Sequence in context: A123137 A014117 A054377 this_sequence A100016 A000610 A023363

Adjacent sequences: A007015 A007016 A007017 this_sequence A007019 A007020 A007021

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

The next term is too large to include.