%I #40 Jan 11 2015 04:33:28

%S 1,2,1,3,10,5,16,8,4,2,1,6,3,10,5,16,8,4,2,1,7,22,11,34,17,52,26,13,

%T 40,20,10,5,16,8,4,2,1,9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,

%U 4,2,1,12,6,3,10,5,16,8,4,2,1

%N Chain of Collatz sequences.

%C The sequence is a chain of Collatz sequences. The first Collatz sequence in the chain is (1). Each of the subsequent Collatz sequences in the chain starts with the minimum positive integer that does not appear in the previous Collatz sequences. If the Collatz conjecture is true, then each Collatz sequence in the chain will end with 1, and the chain will include an infinite number of distinct Collatz sequences. If the Collatz conjecture is false, then the chain will end with the first Collatz sequence that does not converge to 1.

%C T(n,1) = A177729(n). - _Reinhard Zumkeller_, Jan 03 2013

%H Reinhard Zumkeller, <a href="/A192719/b192719.txt">Rows n = 1..120 of triangle, flattened</a>

%H Robert C. Lyons, <a href="/A192719/a192719_2.java.txt">Chain of Collatz sequences generator program in Java</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%e The first Collatz sequence in the chain is (1). The second Collatz sequence in the chain is (2, 1), which starts with 2, since 2 is the smallest positive integer that doesn't appear the first Collatz sequence. The third Collatz sequence in the chain is (3, 10, 5, 16, 8, 4, 2, 1), which starts with 3, since 3 is the smallest positive integer that doesn't appear the previous Collatz sequences.

%e Thus this irregular array starts:

%e 1;

%e 2, 1;

%e 3, 10, 5, 16, 8, 4, 2, 1;

%e 6, 3, 10, 5, 16, 8, 4, 2, 1;

%e 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;

%e 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;

%e ...

%o (Java) See Lyons link.

%o (Haskell)

%o a192719 n k = a192719_tabf !! (n-1) !! (k-1)

%o a192719_row n = a192719_tabf !! (n-1)

%o a192719_tabf = f [1..] where

%o f (x:xs) = (a070165_row x) : f (del xs $ a220237_row x)

%o del us [] = us

%o del us'@(u:us) vs'@(v:vs) | u > v = del us' vs

%o | u < v = u : del us vs'

%o | otherwise = del us vs

%o -- _Reinhard Zumkeller_, Jan 03 2013

%Y Cf. A014682, A070167, A070165, A061641, A177729, A192719.

%Y Cf. A220263 (row lengths); A070165, A220237.

%K nonn,easy,tabf

%O 1,2

%A _Robert C. Lyons_, Dec 31 2012