

A242030


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


3



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
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OFFSET

1,1


COMMENTS

It is conjectured that the trajectory of this Collatzlike 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 Collatzlike 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*x2*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".


LINKS

Table of n, a(n) for n=1..71.


EXAMPLE

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)


PROG

(PARI) {for(n=3, 14, x=n; print1(x, ", "); until(x==8, if(x%2, x=x*31, x=x/2+1); print1(x, ", ")))} \\ Prints flattened triangle.
(PARI) variation(a, b) = {if(!(a==1a==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.


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).
Sequence in context: A093524 A200343 A199456 * A105722 A103559 A021030
Adjacent sequences: A242027 A242028 A242029 * A242031 A242032 A242033


KEYWORD

nonn,tabf


AUTHOR

K. Spage, Aug 11 2014


STATUS

approved



