1,3

This sequence is the unification, in the limit, of the length of Collatz sequences for all fractions whose denominator is odd, and also all integers.

The sequence A210483 gives the triangle read by rows giving trajectory of k/(2n+1) in Collatz problem, k = 1..2n, but particular attention should be paid to numbers in the triangle T(n,k) = (n-k)/(2k+1) for n = 1,2,... and k = 0..n-1.

The example shown below gives a general idea of this regular triangle. This contains all fractions whose denominator is odd and all integers. Now, from T(n,k) we could introduce a 3D triangle in order to produce a complete Collatz sequence starting from each rational T(n,k).

Remark: a(A000124(n)) = A006577(n) because the first column of this triangle generates A006577.

The triangle T(n,k) begins

1;

2, 1/3;

3, 2/3, 1/5;

4, 3/3, 2/5, 1/7;

5, 4/3, 3/5, 2/7, 1/9;

6, 5/3, 4/5, 3/7, 2/9, 1/11;

...

Michel Lagneau, Rows n = 1..100, flattened

J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53.

The triangle of lengths begins

0;

1, 2;

7, 3, 3;

2, 0, 3, 6;

5, 4, 15, 7, 5;

...

Individual numbers have the following Collatz sequences:

[1] => [0] (0 iteration);

[2 1/3] => [1, 2] because: 2 -> 1 => 1 iteration; 1/3 -> 2 -> 1 => 2 iterations;

[3 2/3 1/5] => [7, 3, 3] because: 3->10->5->16->8->4->2->1 => 7 iterations; 2/3 -> 1/3 -> 2 -> 1 => 3 iterations; 1/5 -> 8/5 -> 4/5 -> 2/5 => 3 iterations.

Collatz2[n_] := Module[{lst = NestWhileList[If[EvenQ[Numerator[#]], #/2, 3 # + 1] &, n, Unequal, All]}, If[lst[[-1]] == 1, lst = Drop[lst, -3], If[lst[[-1]] == 2, lst = Drop[lst, -2], If[lst[[-1]] == 4, lst = Drop[lst, -1], If[MemberQ[Rest[lst], lst[[-1]]], lst = Drop[lst, -1]]]]]]; t = Table[s = Collatz2[(n - k)/(2*k + 1)]; Length[s] - 1, {n, 12}, {k, 0, n - 1}]; Flatten[t] (* T. D. Noe, Jan 28 2013 *)

Cf. A210483, A210468, A210688.

Sequence in context: A021370 A248140 A088538 * A257102 A371707 A226626

Adjacent sequences: A210513 A210514 A210515 * A210517 A210518 A210519

nonn,tabl

Michel Lagneau, Jan 26 2013

approved