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k-th term of n in the 3x+1 problem, k >= 1. Square array A(n, k) read by antidiagonals downwards.
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%I #26 Jul 13 2017 17:44:46

%S 4,2,1,1,4,10,4,2,5,2,2,1,16,1,16,1,4,8,4,8,3,4,2,4,2,4,10,22,2,1,2,1,

%T 2,5,11,4,1,4,1,4,1,16,34,2,28,4,2,4,2,4,8,17,1,14,5,2,1,2,1,2,4,52,4,

%U 7,16,34,1,4,1,4,1,2,26,2,22,8,17,6,4,2,4

%N k-th term of n in the 3x+1 problem, k >= 1. Square array A(n, k) read by antidiagonals downwards.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>

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

%F T(m, n) = A(n, m-n+1), 1 <= n <= m, with A(n, k) = C^{[k]}(n), n >= 1, k >= 1, with iterations of the Collatz map C: n -> 3*n+1 if n is odd and n -> n/2 if n is even. - _Wolfdieter Lang_, Jul 13 2017

%e The array A(n, k) starts:

%e n\k | 1 2 3 4 5 6 7 8 9 10 11 12 ...

%e ---------------------------------------------

%e 1 | 4 2 1 4 2 1 4 2 1 4 2 1

%e 2 | 1 4 2 1 4 2 1 4 2 1 4 2

%e 3 | 10 5 16 8 4 2 1 4 2 1 4 2

%e 4 | 2 1 4 2 1 4 2 1 4 2 1 4

%e 5 | 16 8 4 2 1 4 2 1 4 2 1 4

%e 6 | 3 10 5 16 8 4 2 1 4 2 1 4

%e 7 | 22 11 34 17 52 26 13 40 20 10 5 16

%e 8 | 4 2 1 4 2 1 4 2 1 4 2 1

%e 9 | 28 14 7 22 11 34 17 52 26 13 40 20

%e 10 | 5 16 8 4 2 1 4 2 1 4 2 1

%e ...

%e The triangle T(m, n) starts:

%e m\n | 1 2 3 4 5 6 7 8 9 10 ...

%e ------------------------------------

%e 1 | 4

%e 2 | 2 1

%e 3 | 1 4 10

%e 4 | 4 2 5 2

%e 5 | 2 1 16 1 16

%e 6 | 1 4 8 4 8 3

%e 7 | 4 2 4 2 4 10 22

%e 8 | 2 1 2 1 2 5 11 4

%e 9 | 1 4 1 4 1 16 34 2 28

%e 10 | 4 2 4 2 4 8 17 1 14 5

%e ... formatted, _Wolfdieter Lang_, Jul 13 2017

%o (PARI) trajectory(n, terms) = my(i=0, x=n); while(1, if(x%2==0, x=x/2, x=3*x+1); print1(x, ", "); i++; if(i==terms, break))

%o array(n, k) = for(x=1, n, trajectory(x, k); print(""))

%o array(10, 12) \\ print initial 10 rows and 12 columns of array

%Y Rows of A (columns of T): A153727 (row 1), A033478 (row 3), A033479 (row 9).

%Y Columns of A (diagonals of T): A006370 (column 1), A075884 (column 2), A076536 (column 3).

%K nonn,easy,tabl

%O 1,1

%A _Felix Fröhlich_, Jun 10 2017