A355108 -id:A355108 - cp's OEIS frontend

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026

Offset: 0

N. J. A. Sloane and Richard Stanley

Number of binary trees of height less than or equal to n. [Corrected by Orson R. L. Peters, Jan 03 2020]
The rightmost digits cycle (0,1,2,5,6,7,0,1,2,5,6,7,...). - Jonathan Vos Post, Jul 21 2005
Apart from the initial term, a subsequence of A008318. - Reinhard Zumkeller, Jan 17 2008
Partial sums of A001699. - Jonathan Vos Post, Feb 17 2010
Corresponds to the second and second last diagonals of A119687. - John M. Campbell, Jul 25 2011
This is a divisibility sequence. - Michael Somos, Jan 01 2013
Sum_{n>=1} 1/a(n) = 1.739940825174794649210636285335916041018367182486941... . - Vaclav Kotesovec, Jan 30 2015
From Vladimir Vesic, Oct 03 2015: (Start)
Forming Herbrand's domains of formula: (∃x)(∀y)(∀z)(∃k)(P(x)∨Q(y)∧R(k))
where: x->a
       k->f(y,z)
we get:
  H0 = {a}
  H1 = {a, f(a,a)}
  H2 = {a, f(a,a), f(a,f(a,a)), f(f(a,a),a), f(f(a,a),f(a,a))}
  ...
The number of elements in each domain follows this sequence.
(End)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
This is a strong divisibility sequence; see A329429. - Clark Kimberling, Nov 13 2019
From Peter Bala, Oct 31 2022: (Start)
Let k be a positive integer. Clearly, the sequence obtained by reducing a(n) modulo k is eventually periodic. Conjectures:
1) The sequence obtained by reducing a(n) modulo 2^k is eventually periodic with period 2.
2) The sequence obtained by reducing a(n) modulo 10^k is eventually periodic with period 6 (the case k = 1 is noted above).
3) The sequence obtained by reducing a(n) modulo 20^k is eventually periodic with period 6.
4) For n >= floor(k/2) and for 1 <= i <= 6, the value of a(6*n+i) mod 10^k is a constant independent of n. The digits of these 6 constant integers, when read from right to left, are the first k digits of the 10-adic numbers A318135 (i = 1), A318136 (i = 2), A318137 (i = 3), A318138 (i = 4), A318139 (i = 5) and A318140 (i = 6), respectively. An example is given below.
  n      a(6*n+1) mod 10^11
1           10066388901
2           72084948901
3           67988948901
4           61588948901
5           01588948901
6           01588948901
7           01588948901
...             ...
A318135 begins 1, 0, 9, 8, 4, 9, 8, 8, 5, 1, 0, 2, .... (End)

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..13
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!)
A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Transactions (2021) Vol. 1, No. 1, Article 1.
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.
Claudio Gentile, Fabio Vitale, and Anand Rajagopalan, Flattening a Hierarchical Clustering through Active Learning, arXiv:1906.09458 [cs.LG], 2019.
Spencer Hamblen, Rafe Jones, and Kalyani Madhu, The density of primes in orbits of z^d + c, arXiv:1303.6513 [math.NT], 2013; to appear, Int. Math. Res. Not., c. 2015.
Dimitur Krustev, Simple Programs on Binary Trees Testing and Decidable Equivalence, 2016.
Robin Lamarche-Perrin, An Information-theoretic Framework for the Lossy Compression of Link Streams, arXiv:1807.06874 [cs.DS], 2018.
R. Lamarche-Perrin, Y. Demazeau, and J.-M. Vincent, A Generic Algorithmic Framework to Solve Special Versions of the Set Partitioning Problem, Preprint 105, Max-Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig, 2014.
C. Lenormand, Arbres et permutations II, see p. 6.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Michael Penn, a stylish proof that..., YouTube video, 2021.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
M. Tainiter, Algebraic approach to stopping variable problems: Representation theory and applications, J. Combinatorial Theory 9 1970 148-161.
P. Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv preprint arXiv:1507.06944 [cs.LO], 2015.
Stephan Wagner and Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.
Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index to divisibility sequences
Index entries for sequences related to Benford's law

Cf. A001699, A004019, A038044, A056207, A076949, A077496, A091980, A143848.
Cf. A143849, A213437, A247981, A248218, A248219, A318135, A318136, A318137.
Cf. A318138, A318139, A318140, A355108.
Cf. A137560, which enumerates binary trees of height less than n and exactly j leaf nodes. - Robert Munafo, Nov 03 2009

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
end function;
[A003095(n): n in [0..12]]; // G. C. Greubel, Nov 29 2022

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)^2+1) end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 11 2019

NestList[#^2+1&,0,10] (* Harvey P. Dale, Dec 17 2010 *)

{a(n) = if( n<1, 0, 1 + a(n-1)^2)}; /* Michael Somos, Jan 01 2013 */

def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
[A003095(n) for n in range(13)] # G. C. Greubel, Nov 29 2022

a(n) = B_{n-1}(1) where B_n(x) = 1 + x*B_{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... (see A076949). - Benoit Cloitre, 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} A088674(k)/(2*c^(2^n))^(2*k-1). - Gerald McGarvey, Nov 17 2007
a(n) = Sum_{i=1..n} A001699(i). - Jonathan Vos Post, Feb 17 2010
G.f. = x + 2*x^2 + 5*x^3 + 26*x^4 + 677*x^5 + 458330*x^6 + 210066388901*x^7 + ... . - Michael Somos, Jan 01 2013
a(2n) mod 2 = 0 ; a(2n+1) mod 2 = 1. - Altug Alkan, Oct 04 2015
a(n) + a(n-1) = A213437(n). - Peter Bala, Feb 03 2017
0 = a(n)^2*(+a(n+1) + a(n+2)) + a(n+1)^2*(-a(n+1) - a(n+2) - a(n+3)) + a(n+2)^3 for all n>=0. - Michael Somos, Feb 10 2017
a(n) = A091980(2^(n-1)) for n > 0. - Alois P. Heinz, Jul 11 2019

Additional comments from Cyril Banderier, Jun 05 2000
Minor edits by Vaclav Kotesovec, Oct 04 2014
Initial term clarified by Clark Kimberling, Nov 13 2019

A004019 a(0) = 0; for n > 0, a(n) = (a(n-1) + 1)^2.

Original entry on oeis.org

0, 1, 4, 25, 676, 458329, 210066388900, 44127887745906175987801, 1947270476915296449559703445493848930452791204, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352025

Offset: 0

N. J. A. Sloane

Take the standard rooted binary tree of depth n, with 2^(n+1) - 1 labeled nodes. Here is a picture of the tree of depth 3:
             R
            / \
           /   \
          /     \
         /       \
        /         \
       o           o
      / \         / \
     /   \       /   \
    o     o     o     o
   / \   / \   / \   / \
  o   o o   o o   o o   o
Let the number of rooted subtrees be s(n). For example, for n = 1 the s(2) = 4 subtrees are:
  R   R   R      R
     /     \    / \
    o       o  o   o
Then s(n+1) = 1 + 2*s(n) + s(n)^2 = (1+s(n))^2 and so s(n) = a(n+1).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.

Reinhard Zumkeller, Table of n, a(n) for n = 0..11 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Geir Agnarsson, Elie Alhajjar, and Aleyah Dawkins, On locally finite ordered rooted trees and their rooted subtrees, arXiv:2312.11379 [math.CO], 2023.
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!)
Andressa Paola Cordeiro, Alexandre Tavares Baraviera, and Alex Jenaro Becker, Entropy for k-trees defined by k transition matrices, arXiv:2307.05850 [math.DS], 2023. See p. 15.
F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850.
E. Lappo and N. A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
Elmar Teufl and Stephan Wagner, Enumeration problems for classes of self-similar graphs, Journal of Combinatorial Theory, Series A, Volume 114, Issue 7, October 2007, Pages 1254-1277.
Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index entries for sequences related to rooted trees

Cf. A001699, A056207.
Cf. A000290.
Cf. A003095, A091980, A355108.

a004019 n = a004019_list !! n
a004019_list = iterate (a000290 . (+ 1)) 0
-- Reinhard Zumkeller, Feb 01 2013

[n le 1 select 0 else  (Self(n-1)+1)^2: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015

Table[Nest[(1 + #)^2 &, 0, n], {n, 0, 12}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
NestList[(#+1)^2&,0,10] (* Harvey P. Dale, Oct 08 2011 *)

a(n) = if(n==0, 0, (a(n-1) + 1)^2);
vector(20, n, a(n-1)) \\ Altug Alkan, Oct 06 2015

a(n) = A003095(n)^2 = A003095(n+1) - 1 = A056207(n+1) + 1.
It follows from Aho and Sloane that there is a constant c such that a(n) is the nearest integer to c^(2^n). In fact a(n+1) = nearest integer to b^(2^n) - 1 where b = 2.25851845058946539883779624006373187243427469718511465966.... - Henry Bottomley, Aug 30 2005
a(n) is the number of root ancestral configurations for fully symmetric matching gene trees and species trees with 2^n leaves, a(n) = A355108(2^n). - Noah A Rosenberg, Jun 22 2022

One more term from Henry Bottomley, Jul 24 2000

Additional comments from Max Alekseyev, Aug 30 2005

A091980 Recursive sequence; one more than maximum of products of pairs of previous terms with indices summing to current index.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 16, 26, 36, 56, 81, 131, 183, 287, 417, 677, 937, 1457, 2107, 3407, 4759, 7463, 10843, 17603, 24373, 37913, 54838, 88688, 123892, 194300, 282310, 458330, 634350, 986390, 1426440, 2306540, 3221844, 5052452, 7340712, 11917232, 16500522

Offset: 1

Franklin T. Adams-Watters, Mar 15 2004

The maximum is always obtained by taking i as the power of 2 nearest to n/2. - Anna de Mier, Mar 12 2012

a(n) is the number of (binary) max-heaps on n-1 elements from the set {0,1}. a(7) = 16: 000000, 100000, 101000, 101001, 110000, 110010, 110100, 110110, 111000, 111001, 111010, 111011, 111100, 111101, 111110, 111111. - Alois P. Heinz, Jul 09 2019

A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Graphs Combin., 28 (2012), 265-275.

Alois P. Heinz, Table of n, a(n) for n = 1..5652
F. Disanto and N. A. Rosenberg, Enumeration of ancestral configurations for matching gene trees and species trees, J. Comput. Biol. 24 (2017), 831-850. See Section 4.2.
A. de Mier and M. Noy, On the maximum number of cycles in outerplanar and series-parallel graphs, Elect. Notes Discr. Math 34 (2009) 489-493
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Cf. A003095, A056971, A309049.
Partial differences give A168542.
a(n) = A355108(n)+1.
Column k=0 of A370484 and of A372640.

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f->
      1+b(f)*b(n-1-f))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
a:= n-> b(n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2019

a[n_] := a[n] = 1 + Max[Table[a[i] a[n-i], {i, n-1}]]; a[1] = 1;
Array[a, 50] (* Jean-François Alcover, Apr 30 2020 *)

a(n) = 1 + max_{i=1..n-1} a(i)*a(n-i) for n > 1, a(1) = 1.
From Alois P. Heinz, Jul 09 2019: (Start)
a(n) = Sum_{k=0..n-1} A309049(n-1,k).
a(2^(n-1)) = A003095(n). (End)

cp's OEIS Frontend

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

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A004019 a(0) = 0; for n > 0, a(n) = (a(n-1) + 1)^2.

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Haskell

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A091980 Recursive sequence; one more than maximum of products of pairs of previous terms with indices summing to current index.

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Programs

Maple

Mathematica

Formula