A245944 -id:A245944 - cp's OEIS frontend

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

K. Spage, Aug 08 2014

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".

			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)

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).

{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.

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.

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

Original entry on oeis.org

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

K. Spage, Aug 11 2014

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".

			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)

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).

{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.

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.

A245943 Irregular triangle read by rows of a variation of the Collatz iteration with signature (1,2).

Original entry on oeis.org

3, 6, 4, 3, 4, 3, 5, 12, 7, 18, 10, 6, 4, 3, 6, 4, 3, 7, 18, 10, 6, 4, 3, 8, 5, 12, 7, 18, 10, 6, 4, 3, 9, 24, 13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3, 10, 6, 4, 3, 11, 30, 16, 9, 24, 13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3

Offset: 1

K. Spage, Aug 11 2014

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".

			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)

Cf. A245942 for variation type (a=2, b:odd).
Cf. A245944 for variation type (a=2, b:even).
Cf. A242030 for variation type (a=1, b:odd).

{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.

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.

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

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

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A245943 Irregular triangle read by rows of a variation of the Collatz iteration with signature (1,2).

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