OFFSET
1,1
COMMENTS
It is conjectured that the trajectory of this Collatz-like iteration arrives at 3 in a finite number of steps for any initial value x, (x>2). The iterative step is divide by 2 and add 1 if even, or multiply by 3 and subtract 3 if odd. For any initial value the number of steps in the trajectory is the same as the number of steps in the Collatz trajectory starting with (x-2).
This is one of a subset of Collatz-like variations with parameters a = 1 and b = (any positive or negative even integer). The halting value h for type (a=1, b:even) is given by h = a + b. Any odd halting value can be chosen by selecting the appropriate value for b. For any sequence the halting value is arrived at in the same number of steps as the Collatz trajectory starting with (x-b). The iterative function for subset type (a=1, b:even) is x -> (x/2+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 the trajectory of any variation of subset type (a=1, b:even) can be mapped to a Collatz trajectory by b from each element of the trajectory.
The variation with signature type (1,0) belongs to this subset and is in fact the classic Collatz sequence.
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".
EXAMPLE
Some initial rows of the irregular array (r,j):
r: j = (1, 2, 3, ... )
1: (3, 6, 4, 3),
2: (4, 3),
3: (5, 12, 7, 18, 10, 6, 4, 3),
4: (6, 4, 3),
5: (7, 18, 10, 6, 4, 3),
6: (8, 5, 12, 7, 18, 10, 6, 4, 3),
7: (9, 24, 13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3),
8: (10, 6, 4, 3),
9: (11, 30, 16, 9, 24, 13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3),
10: (12, 7, 18, 10, 6, 4, 3),
11: (13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3),
12: (14, 8, 5, 12, 7, 18, 10, 6, 4, 3)
PROG
(PARI) {for(n=3, 14, x=n; print1(x, ", "); until(x==3, if(x%2, x=x*3-3, x=x/2+1); 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(1, 2)} \\ Prints first 12 rows of this irregular array.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
K. Spage, Aug 11 2014
STATUS
approved