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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). 4
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 (list; graph; refs; listen; history; text; internal format)
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]

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.

Sequence in context: A308625 A221469 A117398 * A240852 A309816 A071486

Adjacent sequences:  A295986 A295987 A295988 * A295990 A295991 A295992

KEYWORD

nonn,tabf,look,base

AUTHOR

Rémy Sigrist, Dec 02 2017

STATUS

approved

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Last modified June 22 09:57 EDT 2021. Contains 345375 sequences. (Running on oeis4.)