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A186008 Irregular triangle T(n,k) read by rows, in which row n has the pattern of conjectured dropping times in the Collatz iteration. 1
2, 4, 16, 12, 32, 8, 52, 128, 40, 56, 84, 136, 160, 180, 256, 60, 80, 136, 220, 288, 296, 448, 528, 636, 688, 712, 1024, 152, 232, 384, 648, 704, 788, 856, 1000, 1204, 1416, 1472, 1556, 1592, 1624, 1800, 1972, 2008, 2120, 2356, 2360, 2676, 2744, 2888, 2912, 3064, 3328, 3444, 3680, 3832, 4096 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Consider A126241, the sequence of dropping times in the Collatz iteration. Only zero and the numbers in A020914 can be dropping times. The dropping times in A126241 have a definite pattern. For example, 1 appears at positions n = 2 + 2*i, for i=0,1,2,3,... Similarly, 2 appears at positions n = 5 + 4*i; 4 appears at n = 3 + 16*i; 5 appears at n = 11 + {12,32}*i; and 7 appears at 7 + {8, 52, 128}*i. In general, if we let s=A020914(r) be the r-th possible stopping time, then A126241(n) = s for n = A122442(r) + T(r)*i, where T(r) is the r-th row of this triangle.  The length of row n is A186009(n). The n-th row ends with 2^A020914(n).

The frequency of the r-th dropping time s=A020914(r) can be computed as A186009(r)/2^s. The first few frequencies are 1/2, 1/4, 1/16, 1/16, 3/128, 7/256, 3/256, 15/2048, and 85/8192.

The term "stopping time" is sometimes used instead of "dropping time", but the former usually refers to A006666.

This sequence is closely related to A177789.

REFERENCES

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010. See pp. 33, 35ff.

LINKS

Table of n, a(n) for n=1..57.

Index entries for sequences related to 3x+1 (or Collatz) problem

EXAMPLE

The triangle begins

2

4

16

12, 32

8, 52, 128

40, 56, 84, 136, 160, 180, 256

60, 80, 136, 220, 288, 296, 448, 528, 636, 688, 712, 1024

CROSSREFS

Cf. A014682, A126241, A006666, A186009.

Sequence in context: A073923 A153666 A098819 * A067846 A155893 A196202

Adjacent sequences:  A186005 A186006 A186007 * A186009 A186010 A186011

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, Feb 09 2011

STATUS

approved

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Last modified October 25 03:54 EDT 2021. Contains 348237 sequences. (Running on oeis4.)