cp's OEIS Frontend

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

%I A242030 #12 Oct 03 2015 18:34:37
%S A242030 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,
%T A242030 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,
%U A242030 11,32,17,50,26,14,8,14,8,15,44,23,68,35,104,53,158
%N A242030 Irregular triangle read by rows of a variation of the Collatz iteration with signature (1,1).
%C A242030 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.
%C A242030 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.
%C A242030 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.
%C A242030 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 A242030 Some initial rows of the irregular array (r,j):
%e A242030 r: j = (1, 2, 3, ... )
%e A242030 1: (3, 8),
%e A242030 2: (4, 3, 8),
%e A242030 3: (5, 14, 8),
%e A242030 4: (6, 4, 3, 8),
%e A242030 5: (7, 20, 11, 32, 17, 50, 26, 14, 8),
%e A242030 6: (8, 5, 14, 8),
%e A242030 7: (9, 26, 14, 8),
%e A242030 8: (10, 6, 4, 3, 8),
%e A242030 9: (11, 32, 17, 50, 26, 14, 8),
%e A242030 10: (12, 7, 20, 11, 32, 17, 50, 26, 14, 8),
%e A242030 11: (13, 38, 20, 11, 32, 17, 50, 26, 14, 8),
%e A242030 12: (14, 8)
%o A242030 (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.
%o A242030 (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 A242030 {variation(1,1)} \\ Prints first 12 rows of this irregular array.
%Y A242030 Cf. A245942 for variation type (a=2, b:odd).
%Y A242030 Cf. A245943 for variation type (a=1, b:even).
%Y A242030 Cf. A245944 for variation type (a=2, b:even).
%K A242030 nonn,tabf
%O A242030 1,1
%A A242030 _K. Spage_, Aug 11 2014