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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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).

References

• 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.

Links

• 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

Crossrefs

Cf. A001699, A056207.

Cf. A000290.

Cf. A003095, A091980, A355108.

Programs

Haskell

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

Magma

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

Mathematica

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 *)

PARI

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

Formula

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

Extensions

One more term from Henry Bottomley, Jul 24 2000

Additional comments from Max Alekseyev, Aug 30 2005