cp's OEIS Frontend

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

%I A358551 #11 Nov 22 2022 11:57:52
%S A358551 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,
%T A358551 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,
%U A358551 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A358551 Number of nodes in the ordered rooted tree with binary encoding A014486(n).
%C A358551 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.
%F A358551 a(n) = A072643(n) + 1.
%e A358551 The first few rooted trees in binary encoding are:
%e A358551     0: o
%e A358551     2: (o)
%e A358551    10: (oo)
%e A358551    12: ((o))
%e A358551    42: (ooo)
%e A358551    44: (o(o))
%e A358551    50: ((o)o)
%e A358551    52: ((oo))
%e A358551    56: (((o)))
%e A358551   170: (oooo)
%e A358551   172: (oo(o))
%e A358551   178: (o(o)o)
%e A358551   180: (o(oo))
%e A358551   184: (o((o)))
%t A358551 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]}];
%t A358551 bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
%t A358551 Table[Count[bint[k],_,{0,Infinity}],{k,Select[Range[0,10000],binbalQ]}]
%Y A358551 Run-lengths are A000108.
%Y A358551 Binary encodings are listed by A014486.
%Y A358551 Leaves of the ordered tree are counted by A057514, standard A358371.
%Y A358551 Branches of the ordered tree are counted by A057515.
%Y A358551 Edges of the ordered tree are counted by A072643.
%Y A358551 The Matula-Goebel number of the ordered tree is A127301.
%Y A358551 For standard instead of binary encoding we have A358372.
%Y A358551 The standard ranking of the ordered tree is A358523.
%Y A358551 Depth of the ordered tree is A358550, standard A358379.
%Y A358551 Cf. A000081, A001263, A057122, A358373, A358505, A358524.
%K A358551 nonn
%O A358551 1,2
%A A358551 _Gus Wiseman_, Nov 22 2022