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A295989
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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).
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13
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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
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OFFSET
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0,5
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COMMENTS
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For any n >= 0 and k such that 0 <= k < A001316(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),
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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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]
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MATHEMATICA
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A295989row[n_] := Select[Range[0, n], BitAnd[#, n-#] == 0 &];
Array[A295989row, 25, 0] (* Paolo Xausa, Feb 24 2024 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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