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   A224367 Triangle read by rows giving trajectory of -k/(2n+1) in Collatz problem, k = 1..2n. 0
   0, 1, 2, 4, 5, 7, 6, 9, 10, 11, 11, 13, 12, 4, 5, 1, 6, 3, 2, 4, 7, 8, 9, 8, 10, 11, 9, 13, 11, 13, 12, 14, 15, 5, 16, 16, 6, 18, 17, 20, 17, 19, 7, 4, 5, 4, 6, 1, 5, 6, 7, 7, 2, 9, 6, 8, 7, 17, 18, 9, 19, 9, 10, 20, 20, 11, 10, 22, 11, 24, 21, 23, 21, 36, 37 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,3 COMMENTS Extension of A210483 with negative values, and subset of A224360. LINKS Table of n, a(n) for n=0..74. EXAMPLE The 2nd row [4, 5, 7, 6] gives the number of iterations of -k/5 (the first element is not counted): k=1 => -1/5 ->2/5 -> 1/5 -> 8/5 -> 4/5 with 4 iterations; k=2 => -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 5 iterations; k=3 => -3/5 -> -4/5 -> -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 7 iterations; k=4 => -4/5 -> -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 6 iterations. The array starts: [0]; [1, 2]; [4, 5, 7, 6]; [9, 10, 11, 11, 13, 12]; [4, 5, 1, 6, 3, 2, 4, 7]; ... MATHEMATICA Collatz[n_] := NestWhileList[If[EvenQ[Numerator[-#]], #/2, 3 # + 1] &, n, UnsameQ, All]; t = Join[{{0}}, Table[s = Collatz[-k/(2*n + 1)]; len = Length[s] - 2; If[s[[-1]] == 2, len = len - 1]; len, {n, 10}, {k, 2*n}]]; Flatten[t] (* program from T. D. Noe, adapted for this sequence - see A210483 *). CROSSREFS Cf. A210483, A210468, A224299, A224360. Sequence in context: A026202 A249910 A025511 * A308219 A288758 A210936 Adjacent sequences: A224364 A224365 A224366 * A224368 A224369 A224370 KEYWORD nonn,tabf AUTHOR Michel Lagneau, Apr 05 2013 STATUS approved
  
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  Last modified August 11 17:02 EDT 2024. Contains 375073 sequences. (Running on oeis4.)
explanation: The provided text is an entry from the On-Line Encyclopedia of Integer Sequences (OEIS) describing a specific integer sequence, A224367. It includes a detailed explanation of the sequence, its mathematical properties, and a Mathematica program to generate the sequence. The text does not contain a specific question or a question-and-answer pair.
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: The OEIS is supported by the many generous donors to the OEIS Foundation.
  
   Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
   A224367 Triangle read by rows giving trajectory of -k/(2n+1) in Collatz problem, k = 1..2n. 0
   0, 1, 2, 4, 5, 7, 6, 9, 10, 11, 11, 13, 12, 4, 5, 1, 6, 3, 2, 4, 7, 8, 9, 8, 10, 11, 9, 13, 11, 13, 12, 14, 15, 5, 16, 16, 6, 18, 17, 20, 17, 19, 7, 4, 5, 4, 6, 1, 5, 6, 7, 7, 2, 9, 6, 8, 7, 17, 18, 9, 19, 9, 10, 20, 20, 11, 10, 22, 11, 24, 21, 23, 21, 36, 37 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,3 COMMENTS Extension of A210483 with negative values, and subset of A224360. LINKS Table of n, a(n) for n=0..74. EXAMPLE The 2nd row [4, 5, 7, 6] gives the number of iterations of -k/5 (the first element is not counted): k=1 => -1/5 ->2/5 -> 1/5 -> 8/5 -> 4/5 with 4 iterations; k=2 => -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 5 iterations; k=3 => -3/5 -> -4/5 -> -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 7 iterations; k=4 => -4/5 -> -2/5 -> -1/5 -> 2/5 -> 1/5 -> 8/5 -> 4/5 with 6 iterations. The array starts: [0]; [1, 2]; [4, 5, 7, 6]; [9, 10, 11, 11, 13, 12]; [4, 5, 1, 6, 3, 2, 4, 7]; ... MATHEMATICA Collatz[n_] := NestWhileList[If[EvenQ[Numerator[-#]], #/2, 3 # + 1] &, n, UnsameQ, All]; t = Join[{{0}}, Table[s = Collatz[-k/(2*n + 1)]; len = Length[s] - 2; If[s[[-1]] == 2, len = len - 1]; len, {n, 10}, {k, 2*n}]]; Flatten[t] (* program from T. D. Noe, adapted for this sequence - see A210483 *). CROSSREFS Cf. A210483, A210468, A224299, A224360. Sequence in context: A026202 A249910 A025511 * A308219 A288758 A210936 Adjacent sequences: A224364 A224365 A224366 * A224368 A224369 A224370 KEYWORD nonn,tabf AUTHOR Michel Lagneau, Apr 05 2013 STATUS approved
  
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  Last modified August 11 17:02 EDT 2024. Contains 375073 sequences. (Running on oeis4.)
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distilabel_metadata.raw_output_q_a_detector_0: The provided text is an entry from the On-Line Encyclopedia of Integer Sequences (OEIS) describing a specific integer sequence, A224367. It includes a detailed explanation of the sequence, its mathematical properties, and a Mathematica program to generate the sequence. The text does not contain a specific question or a question-and-answer pair.
  
  Label: no-question