A358551 Number of nodes in the ordered rooted tree with binary encoding A014486(n).

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Author

Gus Wiseman, Nov 22 2022

Keywords

• nonn

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))
		

Crossrefs

Run-lengths are A000108.

Binary encodings are listed by A014486.

Leaves of the ordered tree are counted by A057514, standard A358371.

Branches of the ordered tree are counted by A057515.

Edges of the ordered tree are counted by A072643.

The Matula-Goebel number of the ordered tree is A127301.

For standard instead of binary encoding we have A358372.

The standard ranking of the ordered tree is A358523.

Depth of the ordered tree is A358550, standard A358379.

Cf. A000081, A001263, A057122, A358373, A358505, A358524.

Programs

Mathematica

binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Count[bint[k],_,{0,Infinity}],{k,Select[Range[0,10000],binbalQ]}]

Formula

a(n) = A072643(n) + 1.