A064064 -id:A064064 - cp's OEIS frontend

A064002 List pairs (i,j) with 1 <= i <= j in colexicographic order: (1,1), (1,2), (2,2), (1,3), (2,3), (3,3), (1,4), ... Let a(1) = 1. Then for n>=2 if the (n-1)-st pair is (i,j) then a(n) = a(i) + a(j) + 1.

1, 3, 5, 7, 7, 9, 11, 9, 11, 13, 15, 9, 11, 13, 15, 15, 11, 13, 15, 17, 17, 19, 13, 15, 17, 19, 19, 21, 23, 11, 13, 15, 17, 17, 19, 21, 19, 13, 15, 17, 19, 19, 21, 23, 21, 23, 15, 17, 19, 21, 21, 23, 25, 23, 25, 27, 17, 19, 21, 23, 23, 25, 27, 25, 27, 29, 31, 11, 13, 15, 17, 17

Offset: 1

Claude Lenormand (claude.lenormand(AT)free.fr), Sep 14 2001

All entries are odd. There are A001190(n) occurrences of 2n-1 in this sequence.

a(n) is the number of vertices in the rooted binary tree (every vertex 0 or 2 children) with Colijn-Plazzotta tree number n. - Kevin Ryde, Jul 25 2022

			a(2) = a(1)+a(1)+1 = 3,
a(3) = a(1)+a(2)+1 = 5,
a(4) = a(2)+a(2)+1 = 7,
a(5) = a(1)+a(3)+1 = 7, ...

Kevin Ryde, Table of n, a(n) for n = 1..10000
C. Colijn and G. Plazzotta, A Metric on Phylogenetic Tree Shapes, Systematic Biology, 67 (1) (2018), 113-126.
Kevin Ryde, PARI/GP Code
Wikipedia, Colexicographic order

Cf. A001190, A064064.

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from itertools import count, islice
def bgen(): yield from ((i, j) for j in count(1) for i in range(1, j+1))
def agen():
    a, g = [None, 1], bgen()
    for n in count(2):
        yield a[-1];
        i, j = next(g)
        a.append(a[i] + a[j] + 1)
print(list(islice(agen(), 72))) # Michael S. Branicky, Jul 25 2022

a(n) = 2*A064064(n-1) - 1. - Kevin Ryde, Jul 25 2022

A064065 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 2, 3, 3, 3, 4, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5, 4, 5, 5, 5, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 6, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 4, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 5, 5, 3, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 5, 5

Offset: 0

Henry Bottomley, Aug 31 2001

nonn

Each positive number appears an infinite number of times: e.g., a(k)=1 whenever k-1 is in A006894.

			Start with (0,1). So after initial step have (0, *1*, 0+1 = 1), then (0, 1, *1*, 0+1 = 1, 1+1 = 2), then (0, 1, 1, *1*, 2, 0+1 = 1, 1+1 = 2, 1+1 = 2), then (0, 1, 1, 1, *2*, 1, 2, 2, 0+2 = 2, 1+2 = 3, 1+2 = 3, 1+2 = 3), etc.

Cf. A064064, A064066, A064067.

A064066 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 5, 6, 7, 7, 5, 5, 6, 7, 7, 8, 6, 6, 7, 8, 8, 9, 9, 5, 5, 6, 7, 7, 8, 8, 9, 5, 5, 6, 7, 7, 8, 8, 9, 8, 6, 6, 7, 8, 8, 9, 9, 10, 9, 9, 7, 7, 8, 9, 9, 10, 10, 11, 10, 10, 11, 6, 6, 7, 8, 8, 9, 9, 10, 9, 9, 10, 11, 6, 6, 7, 8, 8, 9, 9, 10, 9, 9, 10, 11, 10, 7, 7, 8, 9, 9

Offset: 0

Henry Bottomley, Aug 31 2001

nonn

			Start with (1,1). So after initial step have (1, *1*, 1+1 = 2), then (1, 1, *2*, 1+2 = 3, 1+2 = 3), then (1, 1, 2, *3*, 3, 1+3 = 4, 1+3 = 4, 2+3 = 5), then (1, 1, 2, 3, *3*, 4, 4, 5, 1+3 = 4, 1+3 = 4, 2+3 = 5, 3+3 = 6), etc.

Each positive number appears A063894 number of times. Cf. A064064, A064065, A064067.

A064067 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 7, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 11, 8, 9, 10, 11, 12, 12, 13, 7, 8, 9, 10, 11, 11, 12, 13, 8, 9, 10, 11, 12, 12, 13, 14, 13, 9, 10, 11, 12, 13, 13, 14, 15, 14, 15, 10, 11, 12, 13, 14, 14, 15, 16, 15, 16, 17, 7, 8, 9, 10, 11, 11, 12, 13, 12, 13, 14

Offset: 0

Henry Bottomley, Aug 31 2001

nonn

			Start with (1,2). So after initial step have (1, *2*, 1+2 = 3), then (1, 2, *3*, 1+3 = 4, 2+3 = 5), then (1, 2, 3, *4*, 5, 1+4 = 5, 2+4 = 6, 3+4 = 7), then (1, 2, 3, 4, *5*, 5, 6, 7, 1+5 = 6, 2+5 = 7, 3+5 = 8, 4+5 = 9), etc.

Each positive number appears A063895 number of times.

Cf. A064064, A064065, A064066.

A120245 a(1) = 1. a(m(m+1)/2 + k) = a(m) + a(k), 1 <= k <= m+1, m >= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 5, 6, 6, 4, 5, 6, 6, 7, 5, 6, 7, 7, 8, 9, 6, 7, 8, 8, 9, 10, 9, 5, 6, 7, 7, 8, 9, 8, 9, 6, 7, 8, 8, 9, 10, 9, 10, 11, 7, 8, 9, 9, 10, 11, 10, 11, 12, 12, 7, 8, 9, 9, 10, 11, 10, 11, 12, 12, 10, 5, 6, 7, 7, 8, 9, 8, 9, 10, 10, 8, 9, 6, 7, 8, 8, 9, 10, 9, 10, 11, 11, 9, 10, 11, 7, 8

Offset: 1

Leroy Quet, Jun 12 2006

Kevin Ryde, Table of n, a(n) for n = 1..10011
Michael De Vlieger, Annotated fan-style heat map of a(n), n = 1..3321, showing n = k(k+1)/2..(k+1)(k+2)/2-1 in semicircular row k, where purples and blues indicate smallest values and reds and magentas largest values in the sequence.
Kevin Ryde, PARI/GP Code

Cf. A120244, A120246, A064064.

nn = 12; a[1] = 1; Do[Set[j, a[m] + a[k]]; Set[a[m (m + 1)/2 + k], j], {m, nn}, {k, m + 1}]; Array[a, # (# + 1)/2] &[nn + 1] (* Michael De Vlieger, Aug 23 2022 *)

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Extended by Ray Chandler, Jun 19 2006

A356917 Irregular triangle read by rows where row n lists the Colijn-Plazzotta subtree numbers, in ascending order, of each vertex of the rooted binary tree with their tree number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 7, 1, 1, 1, 1, 1, 2, 2, 4, 8, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 10, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 11

Offset: 1

Kevin Ryde, Sep 19 2022

Colijn and Plazzotta enumerate rooted binary trees (every vertex 0 or 2 children) by n=1 as a singleton and thereafter tree n is a root with child subtrees x = A002024(n-1) and y = A002260(n-1).
Each row starts with 1's for the childless vertices (A064064(n) of them).
Each row ends with n itself (the tree root).
The second last term in each row is the numerically largest subtree of the root, which is x.
Row lengths are A064002(n), the number of vertices.

			Triangle begins:
      k=1  2  3  4  5  6  7  8  9 10 11
  n=1:  1,
  n=2:  1, 1, 2,
  n=3:  1, 1, 1, 2, 3,
  n=4:  1, 1, 1, 1, 2, 2, 4,
  n=5:  1, 1, 1, 1, 2, 3, 5,
  n=6:  1, 1, 1, 1, 1, 2, 2, 3, 6,
  n=7:  1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 7,
  n=8:  1, 1, 1, 1, 1, 2, 2, 4, 8,
Tree n=6 and its subtree numbers are as follows and row 6 is these subtree numbers in ascending order.
          6  root
        /   \
      3       2
     / \     / \
    2   1   1   1
   / \
  1   1

Kevin Ryde, Table of n, a(n) for rows 1..500, flattened
Caroline Colijn and Giacomo Plazzotta, A Metric on Phylogenetic Tree Shapes, Systematic Biology, volume 67, number 1, January 2018, pages 113-126, see section 2.3 where their L_n = row n here.
Kevin Ryde, PARI/GP Code

Cf. A002024, A002260 (root subtrees).
Cf. A064002 (number of vertices), A064064 (number of childless).
Cf. A356918 (d1 metric).

PARI
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row(n) = sort {row(x), row(y), n} for n>=2, where x = A002024(n-1) and y = A002260(n-1).

cp's OEIS Frontend

A064002 List pairs (i,j) with 1 <= i <= j in colexicographic order: (1,1), (1,2), (2,2), (1,3), (2,3), (3,3), (1,4), ... Let a(1) = 1. Then for n>=2 if the (n-1)-st pair is (i,j) then a(n) = a(i) + a(j) + 1.

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A064065 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=0, a(1)=1.

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A064066 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=1, a(1)=1.

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A064067 n-th step is to add a(n) to each previous number a(k) (excluding itself, i.e., k < n) to produce n more terms of the sequence, starting with a(0)=1, a(1)=2.

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A120245 a(1) = 1. a(m(m+1)/2 + k) = a(m) + a(k), 1 <= k <= m+1, m >= 1.

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A356917 Irregular triangle read by rows where row n lists the Colijn-Plazzotta subtree numbers, in ascending order, of each vertex of the rooted binary tree with their tree number n.

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