A126306 a(n) = number of double-rises (UU-subsequences) in the n-th Dyck path encoded by A014486(n).

0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 4, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3

Offset: 0

Author

Antti Karttunen, Jan 02 2007

Keywords

• nonn

Examples

			A014486(20) = 228 (11100100 in binary), encodes the following Dyck path:
    /\
   /..\/\
  /......\
and there is one rising (left-hand side) slope with length 3 and one with length 1, so in the first slope, consisting of 3 U-steps, there are two cases with two consecutive U-steps (overlapping is allowed), thus a(20)=2.
		

Crossrefs

Cf. A014081, A014486, A000120, A048735, A125976, A057514.

Programs

Python

def ok(n):
    if n==0: return True
    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 False
    return s==0
def a014081(n): return sum(((n>>i)&3==3) for i in range(len(bin(n)[2:]) - 1))
print([a014081(n) for n in range(4001) if ok(n)]) # Indranil Ghosh, Jun 13 2017

Formula

a(n) = A014081(A014486(n)).

a(n) = A000120(A048735(A014486(n))).

a(A125976(n)) = A057514(n)-1, for all n >= 1.