A126303 -id:A126303 - cp's OEIS frontend

A179752 Maximum depth of parenthesizations encoded by A014486, or correspondingly, maximum height for the equivalent general trees.

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

Offset: 0

Antti Karttunen, Aug 03 2010

Each integer n appears first at position given by A014138.

			The terms A014486[1..8] encode the following rooted plane general trees:
.1.......2.......3.......4.......5.......6.......7.......8.
...........................................................
.........................................................o.
.........................................................|.
.................o.................o...o.......o...o.....o.
.................|.................|...|........\./......|.
.o.....o...o.....o.....o.o.o...o...o...o...o.....o.......o.
.|......\./......|......\|/.....\./.....\./......|.......|.
.*.......*.......*.......*.......*.......*.......*.......*.
and the corresponding parenthesizations:
.().....()()....(())...()()()..()(())..(())()..(()())..((()))
thus a(1)=1, a(2)=1, a(3)=2, a(4)=1, a(5)=2, a(6)=2, a(7)=2, a(8)=3.

Antti Karttunen, Table of n, a(n) for n = 0..2055

Cf. A179751, A126303, A126304, A080237, A085197.

blist[m_] := Select[Map[Accumulate, Permutations[PadLeft[Table[1, m], 2*m, -1]]], Min[#] >= 0 &]; Join[{{0}}, Array[Map[Max, blist[#]] &, 6]] (* Paolo Xausa, Mar 04 2024 *)

A126304 a(n) = number of nodes with nonzero even distance to the root in the n-th plane general tree encoded by A014486(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 1, 2, 1, 2

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

			A014486(27) = 696 (1010111000 in binary), encodes the following general plane tree, where the root is marked with * and nodes with even or odd distance to root with 'e's and 'o's, respectively.
.......o
.......|
.......e
.......|
...o.o.o
....\|/.
.....*..
there is one node marked with 'e', thus a(27)=1.

a(n) = A126305(n)-1. Cf. A126303. Scheme-function A014486->parenthesization given in A014486.

(define (A126304 n) (*A126304 (A014486->parenthesization (A014486 n))))
(define (*A126304 s) (cond ((null? s) 0) (else (apply + (map *A126303 s)))))

A126305 a(n) = number of nodes with even distance to the root in the n-th plane general tree encoded by A014486(n). The root node itself is also included.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 2, 3, 2, 3, 3, 4, 3, 3, 4, 4, 5, 4, 3, 4, 3, 4, 2, 3, 3, 4, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 1, 2, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 2, 3, 2, 3, 3, 4, 3, 3, 4, 4, 5, 4, 3, 4, 3, 4, 2, 3, 3, 4, 3, 2, 3, 2, 3

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

a(n) = A126304(n)+1 = A072643(n)-A126303(n). a(n) = A057514(A125982(n)) for all n >=1.

cp's OEIS Frontend

A179752 Maximum depth of parenthesizations encoded by A014486, or correspondingly, maximum height for the equivalent general trees.

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A126304 a(n) = number of nodes with nonzero even distance to the root in the n-th plane general tree encoded by A014486(n).

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Scheme

A126305 a(n) = number of nodes with even distance to the root in the n-th plane general tree encoded by A014486(n). The root node itself is also included.

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Author

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