A224367 Triangle read by rows giving trajectory of -k/(2n+1) in Collatz problem, k = 1..2n.

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

Offset: 0

Author

Michel Lagneau, Apr 05 2013

Keywords

• nonn
• tabf

Comments

Extension of A210483 with negative values, and subset of A224360.

Examples

			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];
  ...
		

Crossrefs

Cf. A210483, A210468, A224299, A224360.

Programs

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 *)