A057514 Number of peaks in mountain ranges encoded by A014486, number of leaves in the corresponding rooted plane trees (the root node is never counted as a leaf).

0, 1, 2, 1, 3, 2, 2, 2, 1, 4, 3, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 2, 1, 5, 4, 4, 4, 3, 4, 3, 4, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 2, 4, 3, 4, 4, 3, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2, 1, 6, 5, 5, 5, 4, 5, 4, 5, 5, 4, 4, 4, 4, 3, 5, 4, 4, 4, 3, 5, 4, 5, 5, 4, 4, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 4, 3, 3, 3, 3

Offset: 0

Author

Antti Karttunen, Sep 03 2000

Keywords

• nonn

Comments

Sum_{i=A014137(n)..(A014137(n+1)-1)} a(i) = A001700(n), i.e., A001700(n) gives the total number of leaves in all ordered trees with n + 1 edges.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..3485
• Antti Karttunen, Gatomorphisms and other excursions ... (Includes Scheme program)
• Antti Karttunen, Newer version of the Scheme code collection

Crossrefs

Cf. A000108, A000120, A001700, A003188, A005811, A014137, A014486, A057515.

a(n)-1 gives the number of zeros in A071153(n) (for n>=1).

Programs

Python

def a005811(n): return bin(n^(n>>1))[2:].count("1")
def ok(n): # This function after Peter Luschny
    B=bin(n)[2:] if n!=0 else 0
    s=0
    for b in B:
        s+=1 if b=="1" else -1
        if s<0: return 0
    return s==0
def A(n): return [0] + [i for i in range(1, n + 1) if ok(i)]
l=A(200)
print([a005811(l[i])//2 for i in range(len(l))]) # Indranil Ghosh, May 21 2017

Formula

a(n) = A005811(A014486(n))/2 = A000120(A003188(A014486(n)))/2.