A245942 Irregular triangle read by rows of a variation of the Collatz iteration with signature (2,3).

5, 10, 7, 16, 6, 5, 10, 7, 16, 7, 16, 8, 6, 5, 10, 7, 16, 9, 22, 13, 34, 19, 52, 28, 16, 10, 7, 16, 11, 28, 16, 12, 8, 6, 5, 10, 7, 16, 13, 34, 19, 52, 28, 16, 14, 9, 22, 13, 34, 19, 52, 28, 16, 15, 40, 22, 13, 34, 19, 52, 28, 16, 16, 10, 7, 16, 17, 46, 25, 70

Offset

1,1

Comments

It is conjectured that the trajectory of this Collatz-like iteration arrives at 16 in a finite number of steps for any initial value x, (x>4). The iterative step is divide by 2 and add 2 if even, or multiply by 3 and subtract 5 if odd. For any odd initial value (x>6) the number of steps in the trajectory is the same as the number of steps in the Collatz trajectory starting with floor(x-3)/2.

This is one of a subset of Collatz-like variations with parameters a = 2 and b = (any positive or negative odd integer). The halting value h for type (a=2, b:odd) is given by h = 12 + b + 1. Any even halting value can be chosen by selecting the appropriate value for b. For any sequence starting with an odd number x, the halting value is arrived at in the same number of steps as the Collatz trajectory starting with floor((x-b)/2). The iterative function for variation type (a=2, b:odd) is x -> (x/2 + ceiling(b/2)) if x is even or x -> (3*x - 2*b + 1) if x is odd.

Two variations belong to the same subset if their (a) parameters are the same and their (b) parameters have the same parity. It is conjectured that any variations belonging to the same subset have equal row lengths. Members of the same subset share other properties. For example, if a trajectory of subset type (a=2, b:odd) starts with an odd number, then the function floor((x-b)/2) maps the element values (x) of this trajectory to a trajectory of A245691.

The subset is part of a wider class of Collatz variations uniquely identified by two parameters (a,b) where a or b can be any integer. The general formula for the halting value is h = 6^(b mod 2)*a + b + b mod 2; the general formula for the iterative mapping function is x -> (x/2 + ceiling(b/2)) if x is even and x -> (3*x - 2*b + a^(a mod 2)) if x is odd. The minimum starting value is b + 1 + b mod 2 for a = 1 or a = 2. Values of a other than 1 or 2 are not always "well behaved".

Links

Table of n, a(n) for n=1..70.

Example

Some initial rows of the irregular array (r,j):
r: j = (1, 2, 3, ... )
1: (5, 10, 7, 16),
2: (6, 5, 10, 7, 16),
3: (7, 16),
4: (8, 6, 5, 10, 7, 16),
5: (9, 22, 13, 34, 19, 52, 28, 16),
6: (10, 7, 16),
7: (11, 28, 16),
8: (12, 8, 6, 5, 10, 7, 16),
8: (13, 34, 19, 52, 28, 16),
10: (14, 9, 22, 13, 34, 19, 52, 28, 16),
11: (15, 40, 22, 13, 34, 19, 52, 28, 16),
12: (16, 10, 7, 16)

Prog

(PARI) {for(n=5, 17, x=n; print1(x, ", "); until(x==16, if(x%2, x=x*3-5, x=x/2+2); print1(x, ", ")))} \\ Prints flattened triangle.
(PARI) variation(a, b) = {if(!(a==1||a==2), print("Enter a=1 or a=2"), h=6^(b%2)*a+b+b%2; c=ceil(b/2); d=2*-b+a^(a%2); for(r=1, 12, x=r+b+b%2; print1(r, ": (", x); until(x==h, if(x%2, x=3*x+d, x=x/2+c); print1(", ", x)); print("), ")))} \\ Generalized version.
{variation(2, 3)} \\ Prints first 12 rows of this irregular array.

Crossrefs

Cf. A245691.

Cf. A245943 for variation type (a=1, b:even).

Cf. A245944 for variation type (a=2, b:even).

Cf. A242030 for variation type (a=1, b:odd).

Sequence in context: A103697 A185341 A067843 * A280943 A316707 A109360

Adjacent sequences: A245939 A245940 A245941 * A245943 A245944 A245945

Keyword

nonn,tabf

Author

K. Spage, Aug 08 2014

Status

approved