cp's OEIS Frontend

From Wolfdieter Lang, Oct 31 2014: (Start)

(old name corrected)

Irregular triangle CT(l, m) such that the first three rows l = 0, 1 and 2 are 1, 2, 4, respectively, and for l >= 3 the row entries CT(l, m) are obtained from replacing the numbers of row l-1 by (2*x-1)/3, 2*x if they are 2 (mod 3) and by 2*x otherwise.

The modified Collatz (or Collatz-Terras) map sends a positive number x to x/2 if it is even and to (3*x+1)/2 if it is odd (see A060322). The present tree (without the complete tree originating at CT(2,1) = 1) can be considered as an incomplete binary tree, with nodes (vertices) of out-degree 2 if they are 2 (mod 3) and out-degree 1 otherwise. In the example below, the edges (branches) could be labeled L (left) or V (vertical).

The row length sequence is A060322(l+1), l>=0. (End)

The Collatz conjecture is true if and only if all odd numbers appear in this sequence.

This sequence is similar to A127824.

For m = 1, the expression for x becomes x = (2^c(1) - 1) / 3.

Also the number of odd x with stopping time n for the Collatz or 3x+1 problem where x->x/2 if x is even, x->(3x+1)/2 if x is odd (see A060322), except that 1 is counted as having stopping time 2 instead of 0.

Equivalently, a(n) is the number of x == 2 (mod 3) with stopping time n-1.

The number of possible c(1),...,c(m) is 2^(n-1) - 2^(n-3); most do not yield integer x.

n-c(m), n-c(m-1), ..., n-c(2) are the stopping times of the odd integers in the Collatz trajectory of x.

For n > 4, a(n) = a(n-2) + a(n-2):(x is 1 mod 6) + a(n-1):(x is 5 mod 6). [I.e., for n > 4, a(n) = a(n-2) + (number of values of x counted in a(n-2) such that x == 1 (mod 6)) + (number of values of x counted in a(n-1) such that x == 5 (mod 6)). - Jon E. Schoenfield, Mar 14 2022]

It is conjectured that lim_{n->oo} a(n)/a(n-1) = 4/3.

With a(2) = 0 this is the first difference sequence of A060322, the row length sequence of A248573 (Collatz-Terras tree). - Wolfdieter Lang, May 04 2015

From Jon E. Schoenfield, Mar 15 2022: (Start)

For n > 4, the set of integers counted in a(n) is the union of three disjoint sets:

(1) the set of integers 4*x+1 obtained using all integers x counted by a(n-2),

(2) the set of integers (4*x-1)/3 obtained using only those integers x counted by a(n-2) that satisfy x == 1 (mod 6), and

(3) the set of integers (2*x-1)/3 obtained using only those integers x counted by a(n-1) that satisfy x == 5 (mod 6). (See Example.) (End)

This is a modified Collatz-Terras map (A060322), called in the Vaillant and Delarue link f.

The Collatz conjecture: iterations of the map f = a: N_0 -> N_0 with n -> a(n) lead always to 0.

The minimal number k with a^{[k]}(n) = 0 is given by A324037(n).

The tree CfTree, related to this map, giving the branches which lead to 0 for each vertex label of level n >= 0 is given in A324246.