A242030 Irregular triangle read by rows of a variation of the Collatz iteration with signature (1,1).

3, 8, 4, 3, 8, 5, 14, 8, 6, 4, 3, 8, 7, 20, 11, 32, 17, 50, 26, 14, 8, 8, 5, 14, 8, 9, 26, 14, 8, 10, 6, 4, 3, 8, 11, 32, 17, 50, 26, 14, 8, 12, 7, 20, 11, 32, 17, 50, 26, 14, 8, 13, 38, 20, 11, 32, 17, 50, 26, 14, 8, 14, 8, 15, 44, 23, 68, 35, 104, 53, 158

Offset: 1

Author

K. Spage, Aug 11 2014

Keywords

• nonn
• tabf

Comments

It is conjectured that the trajectory of this Collatz-like iteration arrives at 8 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 1 if odd.

This is one of a subset of Collatz-like variations with parameters a = 1 and b = (any positive or negative odd integer). The halting value h for type (a=1, b:odd) is given by h = 6 + b + 1. Any odd halting value can be chosen by selecting the appropriate value for b. The iterative function for subset type (a=1, b:odd) is x -> (x/2+ceiling(b/2)) if x is even or x -> (3*x-2*b+1) if x is odd. Unlike the other subsets, there is no simple relationship between the row lengths and the Collatz sequence row lengths.

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.

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".

Examples

			Some initial rows of the irregular array (r,j):
r: j = (1, 2, 3, ... )
1: (3, 8),
2: (4, 3, 8),
3: (5, 14, 8),
4: (6, 4, 3, 8),
5: (7, 20, 11, 32, 17, 50, 26, 14, 8),
6: (8, 5, 14, 8),
7: (9, 26, 14, 8),
8: (10, 6, 4, 3, 8),
9: (11, 32, 17, 50, 26, 14, 8),
10: (12, 7, 20, 11, 32, 17, 50, 26, 14, 8),
11: (13, 38, 20, 11, 32, 17, 50, 26, 14, 8),
12: (14, 8)
		

Crossrefs

Cf. A245942 for variation type (a=2, b:odd).

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

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

Programs

PARI

{for(n=3, 14, x=n; print1(x,", "); until(x==8, if(x%2,x=x*3-1,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(1,1)} \\ Prints first 12 rows of this irregular array.