A295989 Irregular triangle T(n, k), read by rows, n >= 0 and 0 <= k < A001316(n): T(n, k) is the (k+1)-th nonnegative number m such that n AND m = m (where AND denotes the bitwise AND operator).

0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 4, 0, 1, 4, 5, 0, 2, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 1, 2, 3, 8, 9, 10, 11, 0, 4, 8, 12, 0, 1, 4, 5, 8, 9, 12, 13, 0, 2, 4, 6, 8, 10, 12, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0

Offset: 0

Author

Rémy Sigrist, Dec 02 2017

Keywords

• nonn
• tabf
• look
• base

Comments

The (n+1)-th row has A001316(n) terms and sums to n * A001316(n) / 2.

For any n >= 0 and k such that 0 <= k < A001316(n):

- if A000120(n) > 0 then T(n, 1) = A006519(n),

- if A000120(n) > 1 then T(n, 2) = 2^A285099(n),

- if A000120(n) > 0 then T(n, A001316(n)/2 - 1) = A053645(n),

- if A000120(n) > 0 then T(n, A001316(n)/2) = 2^A000523(n),

- if A000120(n) > 0 then T(n, A001316(n) - 2) = A129760(n),

- T(n, A001316(n) - 1) = n,

- the six previous relations correspond respectively (when applicable) to the second term, the third term, the pair of central terms, the penultimate term and the last term of a row,

- T(n, k) AND T(n, A001316(n) - k - 1) = 0,

- T(n, k) + T(n, A001316(n) - k - 1) = n,

- T(n, k) = k for any k < A006519(n+1),

- A000120(T(n, k)) = A000120(k).

If we plot (n, T(n,k)) then we obtain a skewed Sierpinski triangle (see Links section).

If interpreted as a flat sequence a(n) for n >= 0:

- a(n) = 0 iff n = A006046(k) for some k >= 0,

- a(n) = 1 iff n = A006046(2*k + 1) + 1 for some k >= 0,

- a(A006046(k) - 1) = k - 1 for any k > 0.

Examples

			Triangle begins:
  0:   [0]
  1:   [0, 1]
  2:   [0, 2]
  3:   [0, 1, 2, 3]
  4:   [0, 4]
  5:   [0, 1, 4, 5]
  6:   [0, 2, 4, 6]
  7:   [0, 1, 2, 3, 4, 5, 6, 7]
  8:   [0, 8]
  9:   [0, 1, 8, 9]
  10:  [0, 2, 8, 10]
  11:  [0, 1, 2, 3, 8, 9, 10, 11]
  12:  [0, 4, 8, 12]
  13:  [0, 1, 4, 5, 8, 9, 12, 13]
  14:  [0, 2, 4, 6, 8, 10, 12, 14]
  15:  [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
		

Links

• Rémy Sigrist, Rows n = 0..256, flattened
• Rémy Sigrist, Scatterplot of (n, T(n, k)) for n = 0..1023 and k = 0..A001316(n)-1

Crossrefs

Cf. A000120, A000523, A001316 (row lengths), A006046, A006519, A053645, A129760, A285099.

First column of array in A352909.

Programs

Mathematica

A295989row[n_] := Select[Range[0, n], BitAnd[#, n-#] == 0 &];
Array[A295989row, 25, 0] (* Paolo Xausa, Feb 24 2024 *)

PARI

T(n,k) = if (k==0, 0, n%2==0, 2*T(n\2,k), k%2==0, 2*T(n\2, k\2), 2*T(n\2, k\2)+1)

Formula

For any n >= 0 and k such that 0 <= k < A001316(n):

- T(n, 0) = 0,

- T(2*n, k) = 2*T(n, k),

- T(2*n+1, 2*k) = 2*T(n, k),

- T(2*n+1, 2*k+1) = 2*T(n, k) + 1.