A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 35, 37, 41, 43, 64, 128, 129, 137, 139, 163, 169, 171, 256, 257, 293, 512, 515, 529, 547, 553, 555, 641, 649, 651, 675, 681, 683, 1024, 1025, 2048, 2053, 2057, 2059, 2177, 2185, 2187, 2211, 2217, 2219, 2305, 2341, 2563

Offset

1,2

Comments

An ordered tree is balanced if all leaves have the same distance from the root.

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

Links

Table of n, a(n) for n=1..53.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Example

The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  16: (oooo)
  17: ((((o))))
  32: (ooooo)
  35: ((oo)(o))
  37: (((o))((o)))
  41: ((o)(oo))
  43: ((o)(o)(o))

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Select[Range[100], SameQ@@Length/@Position[srt[#], {}]&]

Crossrefs

These trees are counted by A007059.

The unordered version is A184155, counted by A048816.

A000108 counts ordered rooted trees, unordered A000081.

A358379 gives depth of standard ordered trees.

Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

Sequence in context: A351204 A083132 A302602 * A358377 A242175 A325662

Adjacent sequences: A358456 A358457 A358458 * A358460 A358461 A358462

Keyword

nonn

Author

Gus Wiseman, Nov 19 2022

Status

approved