

A007018


a(n) = a(n1)^2 + a(n1), a(0)=1.
(Formerly M1713)


35




OFFSET

0,2


COMMENTS

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 Yshaped tree with 3 edges, we have I, Y, I*I, I*Y, Y*I, Y*Y, where * denotes identification of the roots.  Emeric Deutsch, Oct 31 2002
a(n) has at least n different prime factors. [Saidak]
Subsequence of squarefree numbers (A005117).  Reinhard Zumkeller, Nov 15 2004 [This has been questioned, see MathOverflow link.  Charles R Greathouse IV, Mar 30 2015]
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 "zerosuppressed binary decision diagrams.") For example, a(2)=6 because of the 2variable functions whose truth tables are 1000, 1010, 1011, 1100, 1110, 1111.  Don Knuth, Jun 04 2007
Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429437, 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 Sum_{n>=0} log(1+1/a_n)/2^{n+1}. For example, theta^32  1/2 =~ 3263442.0000000383.  Don Knuth, Jun 04 2007 [Corrected by Darryl K. Nester, Jun 19 2017]
a(n) = A139145(2^(n+1)  1).  Reinhard Zumkeller, 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(n1)) * (A002378(a(n1)) + 1).  Jaroslav Krizek, Sep 13 2009
The next term has 209 digits.  Harvey P. Dale, Sep 07 2011
Urquhart shows that a(n) is the minimum size of a tableau refutation of the clauses of the complete binary tree of depth n, see pp. 432434.  Charles R Greathouse IV, Jan 04 2013
For any positive a(0), the sequence a(n) = a(n1) * (a(n1) + 1) gives a constructive proof that there exists integers with at least n distinct prime factors, e.g. a(n). As a corollary, this gives a constructive proof of Euclid's theorem stating that there are an infinity of primes.  Daniel Forgues, Mar 03 2017
Lower bound for A100016 (with equality for the first 5 terms), where a(n)+1 is replaced by nextprime(a(n)).  M. F. Hasler, May 20 2019


REFERENCES

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 94.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429437.
David Adjiashvili, Sandro Bosio and Robert Weismantel, Dynamic Combinatorial Optimization: a complexity and approximability study, 2012.
Arvind Ayyer, Anne Schilling, Benjamin Steinberg and Nicolas M. ThiĆ©ry, Markov chains, Rtrivial monoids and representation theory, Int. J. Algebra Comput., Vol. 25 (2015), pp. 169231, arXiv preprint, arXiv:1401.4250 [math.CO], 2014.
Umberto Cerruti, Percorsi tra i numeri (in Italian), page 5.
A. Yu. Chirkov, D. V. Gribanov, N. Yu. Zolotykh, On the Proximity of the Optimal Values of the MultiDimensional Knapsack Problem with and without the Cardinality Constraint, arXiv:2004.08589 [math.OC], 2020.
D. R. Curtiss, On Kellogg's Diophantine problem, Amer. Math. Monthly, Vol. 29, No. 10 (1922), pp. 380387.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see p. 577.
MathOverflow, Is OEIS A007018 really a subsequence of squarefree numbers?
Marko R. Riedel, Twocolorings of unordered full binary trees on n levels.
Matthew Roughan, Surreal Birthdays and Their Arithmetic, arXiv:1810.10373 [math.HO], 2018.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Vol. 113, No. 10 (Dec., 2006), pp. 937938.
Alasdair Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, Vol. 1, No. 4 (1995) pp. 425467, esp. p. 434.
Zalman Usiskin, Letter to N. J. A. Sloane, Oct. 1991
Index entries for sequences of form a(n+1)=a(n)^2 + ...


FORMULA

a(n) = A000058(n)1 = A000058(n1)^2  A000058(n1) = 1/(1Sum_{j<n} 1/A000058(j)) where A000058 is Sylvester's sequence.  Henry Bottomley, Jul 23 2001
a(n) = floor(c^(2^n)) where c = A077125 = 1.597910218031873178338070118157...  Benoit Cloitre, Nov 06 2002
a(1)=1, a(n) = Product_{k=1..n1} (a(k)+1).  Benoit Cloitre, Sep 13 2003
If an (additional) initial 1 is inserted, a(n) = Sum_{k<n} a(k)^2.  Franklin T. AdamsWatters, Jun 11 2009
a(n) = A053631(n)/2.  Martin Ettl, Nov 08 2012
Sum_{n>=0} (1)^n/a(n) = A118227.  Amiram Eldar, Oct 29 2020


MAPLE

A007018 := proc(n)
option remember;
local aprev;
if n = 0 then
1;
else
aprev := procname(n1) ;
aprev*(aprev+1) ;
end if;
end proc: # R. J. Mathar, May 06 2016


MATHEMATICA

a=1; lst={}; Do[a=a^2+a; AppendTo[lst, a], {n, 0, 9}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 20 2009 *)
FoldList[#^2 + #1 &, 1, Range@ 8] (* Robert G. Wilson v, Jun 16 2011 *)
NestList[#^2+#&, 1, 10] (* Harvey P. Dale, Sep 07 2011 *)


PROG

(PARI) a(n)=if(n>1, a(n1)+a(n1)^2, n) \\ Edited by M. F. Hasler, May 20 2019
(Maxima)
a[1]:1$
a[n]:=(a[n1] + (a[n1]^2))$
A007018(n):=a[n]$
makelist(A007018(n), n, 1, 10); /* Martin Ettl, Nov 08 2012 */
(Haskell)
a007018 n = a007018_list !! n
a007018_list = iterate a002378 1  Reinhard Zumkeller, Dec 18 2013
(MAGMA) [n eq 1 select 1 else Self(n1)^2 + Self(n1): n in [1..10]]; // Vincenzo Librandi, May 19 2015


CROSSREFS

Cf. A000058, A003687, A004168, A011782, A077125, A117805, A118227.
Lower bound for A100016.
Sequence in context: A054377 A230311 A276416 * A100016 A000610 A023363
Adjacent sequences: A007015 A007016 A007017 * A007019 A007020 A007021


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



