A365991 Square array read by ascending antidiagonals: T(n,k) is the k-th iterate of the 5x+1 function started at n, with n >= 1 and k >= 0.

1, 2, 3, 3, 1, 8, 4, 8, 3, 4, 5, 2, 4, 8, 2, 6, 13, 1, 2, 4, 1, 7, 3, 33, 3, 1, 2, 3, 8, 18, 8, 83, 8, 3, 1, 8, 9, 4, 9, 4, 208, 4, 8, 3, 4, 10, 23, 2, 23, 2, 104, 2, 4, 8, 2, 11, 5, 58, 1, 58, 1, 52, 1, 2, 4, 1, 12, 28, 13, 29, 3, 29, 3, 26, 3, 1, 2, 3

Offset

1,2

Comments

The 5x+1 function (A185452), denoted by T_5(x) in the literature, is defined as T_5(x) = (5x+1)/2 if x is odd, T_5(x) = x/2 if x is even.

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)

Alex V. Kontorovich and Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009, and in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 131-188.

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

Example

The array begins:
  n\k|  0   1   2    3    4    5    6    7    8     9    10    11  ...
  --------------------------------------------------------------------
   1 |  1,  3,  8,   4,   2,   1,   3,   8,   4,    2,    1,    3, ...
   2 |  2,  1,  3,   8,   4,   2,   1,   3,   8,    4,    2,    1, ...
   3 |  3,  8,  4,   2,   1,   3,   8,   4,   2,    1,    3,    8, ...
   4 |  4,  2,  1,   3,   8,   4,   2,   1,   3,    8,    4,    2, ...
   5 |  5, 13, 33,  83, 208, 104,  52,  26,  13,   33,   83,  208, ...
   6 |  6,  3,  8,   4,   2,   1,   3,   8,   4,    2,    1,    3, ...
   7 |  7, 18,  9,  23,  58,  29,  73, 183, 458,  229,  573, 1433, ...
   8 |  8,  4,  2,   1,   3,   8,   4,   2,   1,    3,    8,    4, ...
   9 |  9, 23, 58,  29,  73, 183, 458, 229, 573, 1433, 3583, 8958, ...
  10 | 10,  5, 13,  33,  83, 208, 104,  52,  26,   13,   33,   83, ...
  11 | 11, 28, 14,   7,  18,   9,  23,  58,  29,   73,  183,  458, ...
  12 | 12,  6,  3,   8,   4,   2,   1,   3,   8,    4,    2,    1, ...
  13 | 13, 33, 83, 208, 104,  52,  26,  13,  33,   83,  208,  104, ...
  14 | 14,  7, 18,   9,  23,  58,  29,  73, 183,  458,  229,  573, ...
  15 | 15, 38, 19,  48,  24,  12,   6,   3,   8,    4,    2,    1, ...
  ...

Mathematica

A365991list[dmax_]:=With[{a=Array[NestList[If[OddQ[#], (5#+1)/2, #/2]&, dmax-#, #]&, dmax, 0]}, Array[Diagonal[a, #]&, dmax, 1-dmax]]; A365991list[20] (* Generates 20 antidiagonals *)

Crossrefs

Cf. A185452, A347270, A365484, A365992 (parity), A368301 (main diagonal).

Sequence in context: A236937 A323748 A116155 * A144149 A097005 A068008

Adjacent sequences: A365988 A365989 A365990 * A365992 A365993 A365994

Keyword

nonn,easy,tabl

Author

Paolo Xausa, Sep 25 2023

Status

approved