A157684 -id:A157684 - 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.

A157685 a(n)=#{1<=k<=n : [K(k),K(k+1)]=[2,1]} where K=A000002.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

Benoit Cloitre, Mar 04 2009

nonn

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

Cf. A157684, A157686, A157687

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

a(n)=sum(k=1,n,(1-K(k))*(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

A157654 Triangle T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 4, 0, 4, 1, 1, 6, 2, 2, 6, 1, 1, 8, 4, 0, 4, 8, 1, 1, 10, 6, 2, 2, 6, 10, 1, 1, 12, 8, 4, 0, 4, 8, 12, 1, 1, 14, 10, 6, 2, 2, 6, 10, 14, 1, 1, 16, 12, 8, 4, 0, 4, 8, 12, 16, 1

Offset: 0

Roger L. Bagula, Mar 03 2009

For the cases of m = 0, 1 the triangles reduce to T(n, k, m) = A103451(n, k). - G. C. Greubel, Dec 13 2021

			Triangle begins as:
  1;
  1,  1;
  1,  0,  1;
  1,  2,  2,  1;
  1,  4,  0,  4,  1;
  1,  6,  2,  2,  6,  1;
  1,  8,  4,  0,  4,  8,  1;
  1, 10,  6,  2,  2,  6, 10,  1;
  1, 12,  8,  4,  0,  4,  8, 12,  1;
  1, 14, 10,  6,  2,  2,  6, 10, 14,  1;
  1, 16, 12,  8,  4,  0,  4,  8, 12, 16, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A103451, A244800.

T:= func< n,k,q | k eq 0 or k eq n select 1 else q*Abs( (n-k)^(q-1) - k^(q-1) ) >;
[T(n,k,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 13 2021

T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, m*Abs[(n-k)^(m-1) - k^(m-1)]];
Table[T[n,k,2], {n,0,15}, {k,0,n}]//Flatten

def A157684(n,k,q): return 1 if (k==0 or k==n) else q*abs((n-k)^(q-1) - k^(q-1))
flatten([[A157684(n,k,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021

T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2.
From G. C. Greubel, Dec 13 2021: (Start)
Sum_{k=0..n} T(n, k, 2) = (-1)*[n==0] + A244800(n-1).
T(2*n, n, 2) = A000007(n). (End)

Edited by G. C. Greubel, Dec 13 2021

cp's OEIS Frontend

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

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A157685 a(n)=#{1<=k<=n : [K(k),K(k+1)]=[2,1]} where K=A000002.

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

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A157654 Triangle T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2, read by rows.

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