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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). 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partial sums from A014137[i]th to A014137[i+1]-1:th term of this sequence produce central binomial coefficients C(2n+1,n+1) (see comment at A001700): [seq(add(A057514[j],j=CatPsum(i)..(CatPsum(i+1)-1)),i=0..upto_n)];

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..3485

A. Karttunen, Gatomorphisms and other excursions ... (Includes Scheme program)

FORMULA

a(n) = wt(GrayCode(A014486[n]))/2 = A000120[A003188[A014486[n]]]/2 = A005811[A014486[n]]/2

MAPLE

Cat := n -> binomial(2*n, n)/(n+1); CatPsum := proc(n) option remember; if(0 = n) then RETURN(1); else RETURN(Cat(n)+CatPsum(n-1)); fi; end;

PROG

(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 xrange(1, n + 1) if ok(i)]

l=A(200000)

print [a005811(l[i])/2 for i in xrange(len(l))] # Indranil Ghosh, May 21 2017

CROSSREFS

Cf. A057515. For Maple procedure GrayCode see A055095. a(n)-1 gives the number of zeros in A071153(n) (for n>=1).

Sequence in context: A262324 A286364 A084216 * A273568 A140720 A033559

Adjacent sequences:  A057511 A057512 A057513 * A057515 A057516 A057517

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 03 2000

STATUS

approved

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Last modified February 24 03:21 EST 2018. Contains 299595 sequences. (Running on oeis4.)