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A358459
Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).
1
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.
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.
Sequence in context: A351204 A083132 A302602 * A358377 A242175 A325662
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2022
STATUS
approved