A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

1, 4, 8, 16, 18, 25, 32, 36, 50, 57, 64, 72, 100, 114, 121, 128, 137, 144, 200, 228, 242, 249, 256, 258, 274, 281, 288, 385, 393, 400, 456, 484, 498, 505, 512, 516, 548, 562, 569, 576, 770, 786, 793, 800, 897, 905, 912, 968, 996, 1010, 1017, 1024, 1032, 1096

Offset

1,2

Comments

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..54.

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

Example

The initial terms and their corresponding trees:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   18: ((oo)o)
   25: (o(oo))
   32: (ooooo)
   36: ((oo)oo)
   50: (o(oo)o)
   57: (oo(oo))
   64: (oooooo)
   72: ((oo)ooo)
  100: (o(oo)oo)
  114: (oo(oo)o)
  121: (ooo(oo))
  128: (ooooooo)
  137: ((oo)(oo))
  144: ((oo)oooo)
  200: (o(oo)ooo)

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], FreeQ[srt[#], _[__]?(Length[#]==1&)]&]

Crossrefs

These trees are counted by A005043.

The series-reduced case appears to be counted by A284778.

The unordered version is A291636, counted by A001678.

A000081 counts unlabeled rooted trees, ranked by A358378.

A358371 and A358372 count leaves and nodes in standard ordered rooted trees.

A358374 ranks ordered identity trees, counted by A032027.

A358375 ranks ordered binary trees, counted by A126120.

Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.

Sequence in context: A312774 A312775 A312776 * A312777 A312778 A312779

Adjacent sequences: A358373 A358374 A358375 * A358377 A358378 A358379

Keyword

nonn

Author

Gus Wiseman, Nov 14 2022

Status

approved