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

0,5

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.

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

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.

Example

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]

Mathematica

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

Prog

(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)

Crossrefs

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

First column of array in A352909.

Sequence in context: A221469 A350369 A117398 * A240852 A363612 A309816

Adjacent sequences: A295986 A295987 A295988 * A295990 A295991 A295992

Keyword

nonn,tabf,look,base

Author

Rémy Sigrist, Dec 02 2017

Status

approved