A004168 -id:A004168 - 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

A082732 a(1) = 1, a(2) = 3, a(n) = LCM of all the previous terms + 1.

Original entry on oeis.org

1, 3, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693

Offset: 1

Amarnath Murthy, Apr 14 2003

nonn

The LCM is in fact the product of all previous terms. From a(5) onwards the terms alternately end in 57 and 93.

Cf. A000058, A004168, A144743, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

a[1] = 1; a[2] = 3; a[n_] := Apply[LCM, Table[a[i], {i, 1, n - 1}]] + 1; Table[ a[n], {n, 1, 10}]
c=1.8806785436830780944921917650127503562630617563236301969047995953391479871\
7695395204087358090874194124503892563356447954254847544689332763; Table[c^(2^n),{n,1,6}] or a = {}; k = 4; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a (* Artur Jasinski, Sep 22 2008 *)

For n>=3, a(n+1) = a(n)^2 - a(n) + 1.
For n>=3, a(n) = A004168(n-3) + 1. - Max Alekseyev, Aug 09 2019
1/3 = Sum_{n=3..oo} 1/a(n) = 1/4 + 1/13 + 1/157 + 1/24493 + ... or 1 = Sum_{n=3..oo} 3/a(n) = 3/4 + 3/13 + 3/157 + 3/24493 + .... If we take segment of length 1 and cut off in each step fragment of maximal length such that numerator of fraction is 3, denominators of such fractions will be successive numbers of this sequence. - Artur Jasinski, Sep 22 2008
a(n+2)=1.8806785436830780944921917650127503562630617563236301969047995953391\
4798717695395204087358090874194124503892563356447954254847544689332763...^(2^n). -  Artur Jasinski, Sep 22 2008

More terms from Robert G. Wilson v, Apr 15 2003

A376062 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence {7/6, 5/4, 5/4, 5/4, ...}.

Original entry on oeis.org

2, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693

Offset: 1

N. J. A. Sloane, Sep 14 2024

nonn

This sequence and A376186 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.

This is essentially the same sequence as A004168 and A082732.

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]

Cf. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376061, A376185.

Join[{2}, RecurrenceTable[{a[n+1] == a[n]^2 - a[n] + 1, a[2] == 4}, a, {n, 2, 9}]] (* Amiram Eldar, Sep 15 2024 *)

a(n+1) = a(n)^2 - a(n) + 1 for n >= 2.

A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).

Original entry on oeis.org

2, 5, 17, 341, 92753, 10753782821, 92515075960384748177, 10698799099944699918936107506299150093941, 91571441744782016867976366392607084634231243149599342901251284090792487979854033

Offset: 1

N. J. A. Sloane, Sep 15 2024

This sequence and A376062 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...

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]

Cf. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376062, A376186.

A376186 a(n) = denominator of the sum S(n) defined in A376184.

Original entry on oeis.org

8, 40, 680, 231880, 21507565640, 231287689900961870440, 21397598199889399837872215012598300187880, 228928604361955042169940915981517711585578107873998357253128210226981219949635080

Offset: 1

N. J. A. Sloane, Sep 15 2024

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...

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]

Cf. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376062, A376184, A376185.

A298484 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is defined by the following: A(1,k) = k and A(n,k) = A(n-1,k)*(A(n-1,k)+1) for n > 1.

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 4, 12, 42, 42, 5, 20, 156, 1806, 1806, 6, 30, 420, 24492, 3263442, 3263442, 7, 42, 930, 176820, 599882556, 10650056950806, 10650056950806, 8, 56, 1806, 865830, 31265489220, 359859081592975692, 113423713055421844361000442, 113423713055421844361000442

Offset: 1

Seiichi Manyama, Jan 20 2018

			Square array begins:
      1,       2,         3,           4,           5,              6, ...
      2,       6,        12,          20,          30,             42, ...
      6,      42,       156,         420,         930,           1806, ...
     42,    1806,     24492,      176820,      865830,        3263442, ...
   1806, 3263442, 599882556, 31265489220, 74966245730, 10650056950806, ...

Seiichi Manyama, Antidiagonals n = 1..13, flattened

Columns k=1..3 give A007018(n-1), A007018, A004168(n-1).

Rows n=1..3 give A000027, A002378, A169938.

A[1, k_?Positive] := k; A[n_?Positive, k_?Positive] := A[n, k] = A[n - 1, k]*(A[n - 1, k] + 1); Table[A[n - k + 1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Feb 05 2018 *)

cp's OEIS Frontend

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

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A082732 a(1) = 1, a(2) = 3, a(n) = LCM of all the previous terms + 1.

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A376062 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence {7/6, 5/4, 5/4, 5/4, ...}.

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A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2*i)=3/2, b(2*i+1)=6/5 (i>0).

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A376186 a(n) = denominator of the sum S(n) defined in A376184.

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A298484 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is defined by the following: A(1,k) = k and A(n,k) = A(n-1,k)*(A(n-1,k)+1) for n > 1.

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A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).