A157685 -id:A157685 - cp's OEIS frontend

A157686 a(n) = A157684(n) - A157685(n).

1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0

Offset: 1

Benoit Cloitre, Mar 04 2009

nonn

For n>=1 a(n) = A000002(n+1)-1 and also a(n) = A123594(n+3).

Rabie A. Mahmoud, Hardware Implementation of Binary Kolakoski Sequence, Research Gate, 2015.

Cf. A000002, A123594, A157684, A157685, A157687.

Partial sums of A054354.

A157684 a(n)=#{1<=k<=n : [K(k),K(k+1)]=[1,2]} where K=A000002.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27

Offset: 1

Benoit Cloitre, Mar 04 2009

nonn

Presumably a(n)=n/3+o(n)

Cf. A157685, A157686, A157687

a(n)=sum(k=1,n,if(abs(A000002(k)-1)+abs(A000002(k+1)-2),0,1))

a(n)=sum(k=1,n,(K(k+1)-1)*(K(k+1)-K(k))) where K(k)=A000002(k).

A157687 a(n) = n - A054353(A156351(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 1

Benoit Cloitre, Mar 04 2009

nonn

Cf. A156351, A156728, A157684, A157685, A157686

a(n) = 1 iff n is in A078649, a(n) = 0 iff n is in A054353.
a(n) = n-A054353(A157684(n)+A157685(n)).
a(n) = 1 - A156728(n). - Alan Michael Gómez Calderón, Dec 19 2024

cp's OEIS Frontend

A157686 a(n) = A157684(n) - A157685(n).

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A157684 a(n)=#{1<=k<=n : [K(k),K(k+1)]=[1,2]} where K=A000002.

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A157687 a(n) = n - A054353(A156351(n)).

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