A108225 -id:A108225 - cp's OEIS frontend

A109070 Number of digits in numbers appearing in A108225.

1, 1, 1, 1, 2, 2, 4, 7, 13, 25, 50, 99, 196, 392, 783, 1566, 3131, 6261, 12521, 25042, 50084, 100166, 200332, 400663, 801325, 1602649, 3205298, 6410595, 12821190, 25642379, 51284758

Offset: 0

Zak Seidov, Jun 18 2005

Starting with a(14)=783, a(n) = 2*a(n-1) - (1 or 0).

			A109070(10)=50 because A108225(10)=16217557574922386301420536972254869595782763547562 has 50 digits.

Cf. A108225.

nxt[{a_,b_}]:={b,((a+b)(b-a+1))/2}; Join[{1},IntegerLength/@ Transpose[ NestList[nxt,{0,2},31]][[2]]] (* Harvey P. Dale, Feb 14 2012 *)

A007501 a(0) = 2; for n >= 0, a(n+1) = a(n)*(a(n)+1)/2.

Original entry on oeis.org

2, 3, 6, 21, 231, 26796, 359026206, 64449908476890321, 2076895351339769460477611370186681, 2156747150208372213435450937462082366919951682912789656986079991221

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Number of nonisomorphic complete binary trees with leaves colored using two colors. - Brendan McKay, Feb 01 2001

With a(0) = 2, a(n+1) is the number of possible distinct sums between any number of elements in {1,...,a(n)}. - Derek Orr, Dec 13 2014

			Example for depth 2 (the nonisomorphic possibilities are AAAA, AAAB, AABB, ABAB, ABBB, BBBB):
         o
        / \
       /   \
      o     o
     / \   / \
    /   \ /   \
    A   B B   B

W. H. Cutler, Subdividing a Box into Completely Incongruent Boxes, J. Rec. Math., 12 (1979), 104-111.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..12
G. L. Honaker, Jr., 41041 (another Prime Pages' Curiosity)
J. C. Kieffer, Hierarchical Type Classes and Their Entropy Functions, in 2011 First International Conference on Data Compression, Communications and Processing, pp. 246-254; Digital Object Identifier: 10.1109/CCP.2011.36.
J. V. Post, Math Pages [wayback copy]
J.S. Seneschal, Iteration of Complete Graphs
Stephan Wagner, Enumeration of highly balanced trees

Cf. A000217, A006893.
Cf. A117872 (parity), A275342 (2-adic valuation).
Cf. A129440.
Cf. A013589 (start=4), A050542 (start=5), A050548 (start=7), A050536 (start=8), A050909 (start=9).
Cf.  A108225, A145272.

a007501 n = a007501_list !! n
a007501_list = iterate a000217 2  -- Reinhard Zumkeller, Aug 15 2013

f[n_Integer] := n(n + 1)/2; NestList[f, 2, 10]

PARI
```
a(n)=if(n<1,2,a(n-1)*(1+a(n-1))/2)
```

a(n) = A006893(n+1) + 1.
a(n+1) = A000217(a(n)). - Reinhard Zumkeller, Aug 15 2013
a(n) ~ 2 * c^(2^n), where c = 1.34576817070125852633753712522207761954658547520962441996... . - Vaclav Kotesovec, Dec 17 2014
a(n) = A145272(n) + a(n-1). - J.S. Seneschal, Jul 17 2025

A354970 Smallest value of the Colijn-Plazzotta rank among n-leaved unlabeled binary rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 11, 20, 22, 28, 29, 53, 56, 66, 67, 202, 211, 252, 254, 401, 407, 435, 436, 1408, 1432, 1594, 1597, 2202, 2212, 2278, 2279, 20369, 20504, 22358, 22367, 31838, 31879, 32384, 32386, 80455, 80602, 83023, 83029, 94803, 94831, 95266, 95267

Offset: 1

Noah A Rosenberg, Jun 14 2022

nonn

a(n) gives the rank of the unlabeled binary rooted tree, among those with n leaves, that has the smallest rank according to the bijection of Colijn and Plazzotta (2018) between unlabeled binary rooted trees and positive integers.

C. Colijn and G. Plazzotta, A metric on phylogenetic tree shapes, Syst. Biol., 67 (2018), 113-126.
Michael R. Doboli, Hsien-Kuei Hwang, and Noah A. Rosenberg, Periodic Behavior of the Minimal Colijn-Plazzotta Rank for Trees with a Fixed Number of Leaves, In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 18:1-18:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2024).
N. A. Rosenberg, On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees, Discr. Appl. Math., 291 (2021), 88-98.
Kevin Ryde, PARI/GP Code

Cf. A001190, A108225.

a := proc(n) option remember; local r, h; if n = 1 then 1 else
r := irem(n, 2); h := a((n + r)/2); (h^2 - h)/2 + a((n - r)/2) + 1 fi end:
seq(a(n), n = 1..48); # Peter Luschny, Jun 15 2022

a [n_] := a[n] = Module[{r, h}, If[ n == 1, 1, r = Mod[n, 2]; h = a[(n+r)/2]; (h^2-h)/2 + a[(n-r)/2] + 1]];
Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Nov 08 2023, after Peter Luschny *)

PARI
```
\\ See links.
```

from collections import deque
from itertools import islice
def A354970_gen(): # generator of terms
    aqueue, f, a, b = deque([]), False, 1, 2
    yield from (1,2)
    while True:
        c = b*(b-1)//2+1+(b if f else a)
        yield c
        aqueue.append(c)
        if f: a, b = b, aqueue.popleft()
        f = not f
A354970_list = list(islice(A354970_gen(),40)) # Chai Wah Wu, Jun 17 2022

a(n) = a(n/2)*(a(n/2)-1)/2 + 1 + a(n/2) for even n; a(n) = a((n+1)/2)*(a((n+1)/2)-1)/2 + 1 + a((n-1)/2) for odd n>1; a(1)=1.

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A109070 Number of digits in numbers appearing in A108225.

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A007501 a(0) = 2; for n >= 0, a(n+1) = a(n)*(a(n)+1)/2.

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A354970 Smallest value of the Colijn-Plazzotta rank among n-leaved unlabeled binary rooted trees.

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