A003687 -id:A003687 - cp's OEIS frontend

A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1.

1, 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806

Offset: 0

N. J. A. Sloane

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, Oct 31 2002
Equivalently, the number of acyclic digraphs (dags) that unravel to a perfect binary tree of height n. - Nachum Dershowitz, Jul 03 2022
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 "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. - Don 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 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]
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. 432-434. - Charles R Greathouse IV, Jan 04 2013
For any positive a(0), the sequence a(n) = a(n-1) * (a(n-1) + 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

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).

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, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
David Adjiashvili, Sandro Bosio and Robert Weismantel, Dynamic Combinatorial Optimization: a complexity and approximability study, 2012.
Gilles Audemard, Steve Bellart, Louenas Bounia, Frédéric Koriche, Jean-Marie Lagniez, and Pierre Marquis, On the Explanatory Power of Decision Trees, arXiv:2108.05266 [cs.AI], 2021.
Arvind Ayyer, Anne Schilling, Benjamin Steinberg and Nicolas M. Thiéry, Markov chains, R-trivial monoids and representation theory, Int. J. Algebra Comput., Vol. 25 (2015), pp. 169-231, arXiv preprint, arXiv:1401.4250 [math.CO], 2014.
Umberto Cerruti, Percorsi tra i numeri (in Italian), page 5.
A. Yu. Chirkov, D. V. Gribanov and N. Yu. Zolotykh, On the Proximity of the Optimal Values of the Multi-Dimensional 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. 380-387.
Christian Elsholtz and Stefan Planitzer, Sums of four and more unit fractions and approximate parametrizations, arXiv:2012.05984 [math.NT], 2020.
Steven Finch, Exercises in Iterational Asymptotics, arXiv:2411.16062 [math.NT], 2024. See p. 9.
Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see p. 577.
Diana Maimuţ and George Teşeleanu, Inferring Bivariate Polynomials for Homomorphic Encryption Application, Cryptology ePrint Archive (2023) Art. 844. See p. 16.
MathOverflow, Is OEIS A007018 really a subsequence of squarefree numbers?.
Marko R. Riedel, Two-colorings 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. 937-938.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
Bertrand Teguia Tabuguia, Computing with D-Algebraic Sequences, arXiv:2412.20630 [math.AG], 2024. See p. 9.
Alasdair Urquhart, The complexity of propositional proofs, Bull. Symbolic Logic, Vol. 1, No. 4 (1995) pp. 425-467, 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 + ....

Cf. A000058, A003687, A004168, A011782, A077125, A117805, A118227, A371321.
Lower bound for A100016.
Row sums of A122888.

a007018 n = a007018_list !! n
a007018_list = iterate a002378 1  -- Reinhard Zumkeller, Dec 18 2013

[n eq 1 select 1 else Self(n-1)^2 + Self(n-1): n in [1..10]]; // Vincenzo Librandi, May 19 2015

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

FoldList[#^2 + #1 &, 1, Range@ 8] (* Robert G. Wilson v, Jun 16 2011 *)
NestList[#^2 + #&, 1, 10] (* Harvey P. Dale, Sep 07 2011 *)

a[1]:1$
a[n]:=(a[n-1] + (a[n-1]^2))$
A007018(n):=a[n]$
makelist(A007018(n),n,1,10); /* Martin Ettl, Nov 08 2012 */

a(n)=if(n>0,my(x=a(n-1));x^2+x,1) \\ Edited by M. F. Hasler, May 20 2019 and Jason Yuen, Mar 01 2025

from itertools import islice
def A007018_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a *= a+1
A007018_list = list(islice(A007018_gen(),9)) # Chai Wah Wu, Mar 19 2024

a(n) = A000058(n)-1 = A000058(n-1)^2 - A000058(n-1) = 1/(1-Sum_{jA000058(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..n-1} (a(k)+1). - Benoit Cloitre, Sep 13 2003
a(n) = A139145(2^(n+1) - 1). - Reinhard Zumkeller, Apr 10 2008
If an (additional) initial 1 is inserted, a(n) = Sum_{kFranklin T. Adams-Watters, Jun 11 2009
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). - Jaroslav Krizek, Sep 13 2009
a(n) = A053631(n)/2. - Martin Ettl, Nov 08 2012
Sum_{n>=0} (-1)^n/a(n) = A118227. - Amiram Eldar, Oct 29 2020
Sum_{n>=0} 1/a(n) = A371321. - Amiram Eldar, Mar 19 2024

A081477 Complement of A086377.

Original entry on oeis.org

2, 3, 5, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 44, 46, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 65, 67, 68, 70, 72, 73, 75, 77, 79, 80, 82, 84, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 102, 104, 106, 108, 109, 111, 113, 114, 116, 118

Offset: 1

N. J. A. Sloane, Oct 12 2008

nonn

The old entry with this sequence number was a duplicate of A003687.
Is A086377 the sequence of positions of 1 in A189687? - Clark Kimberling, Apr 25 2011
The answer to Kimberling's question is: yes. See the Bosma-Dekking-Steiner paper. - Michel Dekking, Oct 14 2018

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Wieb Bosma, Michel Dekking, Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), arXiv 1710.01498 math.NT (2018).
Wieb Bosma, Michel Dekking, Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A4.

Cf. A086377, A004641, A189687.

t = Nest[Flatten[# /. {0->{0,1,1}, 1->{0,1}}] &, {0}, 5] (*A189687*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (* A086377 conjectured *)
Flatten[Position[t, 1]] (* A081477 conjectured *)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189688*)
(* Clark Kimberling, Apr 25 2011 *)

Conjectures from Clark Kimberling, Aug 03 2022: (Start)
[a(n)*r] = n + [n*r] for n >= 1, where r = sqrt(2) and [ ] = floor.
{a(n)*sqrt(2)} > 1/2 if n is in A120753, where { } = fractional part; otherwise n is in A120752. (End)

Name corrected by Michel Dekking, Jan 04 2019

A081478 Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the denominators.

Original entry on oeis.org

1, 2, 2, -2, 6, -6, 42, -42, 1806, -1806, 3263442, -3263442, 10650056950806, -10650056950806, 113423713055421844361000442, -113423713055421844361000442, 12864938683278671740537145998360961546653259485195806

Offset: 1

Amarnath Murthy, Mar 24 2003

The mapping f(a/b) = (a + b)/(a - b). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the periodic sequence 2/1,3/1,2/1,3/1,...

Seiichi Manyama, Table of n, a(n) for n = 1..26

A003687 gives the numerators.

Cf. A007018.

Last /@ NestList[{(#1 - #2), #1 #2} & @@ # &, {2, 1}, 16] (* Michael De Vlieger, Sep 04 2016 *)

# Variant with first four terms slightly different. Absolute values.
def A081478_abs():
    x, y = 1, 2
    yield x
    while True:
       yield x
       x, y = x * y, x//y + 1
a = A081478_abs(); print([next(a) for i in range(17)])  # Peter Luschny, Dec 17 2015

a(2n-1) = A007018(n-1), a(2n) = -A007018(n-1) for n >= 2. - Jianing Song, Oct 10 2021

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

cp's OEIS Frontend

A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Python

Formula

A081477 Complement of A086377.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A081478 Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the denominators.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions