cp's OEIS Frontend

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

%I A245943 #10 Oct 03 2015 18:35:18
%S A245943 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,
%T A245943 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,
%U A245943 9,24,13,36,19,54,28,15,42,22,12,7,18,10,6,4,3
%N A245943 Irregular triangle read by rows of a variation of the Collatz iteration with signature (1,2).
%C A245943 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).
%C A245943 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.
%C A245943 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.
%C A245943 The variation with signature type (1,0) belongs to this subset and is in fact the classic Collatz sequence.
%C A245943 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".
%e A245943 Some initial rows of the irregular array (r,j):
%e A245943 r: j = (1, 2, 3, ... )
%e A245943 1: (3, 6, 4, 3),
%e A245943 2: (4, 3),
%e A245943 3: (5, 12, 7, 18, 10, 6, 4, 3),
%e A245943 4: (6, 4, 3),
%e A245943 5: (7, 18, 10, 6, 4, 3),
%e A245943 6: (8, 5, 12, 7, 18, 10, 6, 4, 3),
%e A245943 7: (9, 24, 13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3),
%e A245943 8: (10, 6, 4, 3),
%e A245943 9: (11, 30, 16, 9, 24, 13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3),
%e A245943 10: (12, 7, 18, 10, 6, 4, 3),
%e A245943 11: (13, 36, 19, 54, 28, 15, 42, 22, 12, 7, 18, 10, 6, 4, 3),
%e A245943 12: (14, 8, 5, 12, 7, 18, 10, 6, 4, 3)
%o A245943 (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.
%o A245943 (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.
%o A245943 {variation(1,2)} \\ Prints first 12 rows of this irregular array.
%Y A245943 Cf. A245942 for variation type (a=2, b:odd).
%Y A245943 Cf. A245944 for variation type (a=2, b:even).
%Y A245943 Cf. A242030 for variation type (a=1, b:odd).
%K A245943 nonn,tabf
%O A245943 1,1
%A A245943 _K. Spage_, Aug 11 2014