A365484 Square array read by ascending antidiagonals: T(n,k) is the k-th iterate of the 3x+1 function started at n, with n >= 1 and k >= 0.

1, 2, 2, 3, 1, 1, 4, 5, 2, 2, 5, 2, 8, 1, 1, 6, 8, 1, 4, 2, 2, 7, 3, 4, 2, 2, 1, 1, 8, 11, 5, 2, 1, 1, 2, 2, 9, 4, 17, 8, 1, 2, 2, 1, 1, 10, 14, 2, 26, 4, 2, 1, 1, 2, 2, 11, 5, 7, 1, 13, 2, 1, 2, 2, 1, 1, 12, 17, 8, 11, 2, 20, 1, 2, 1, 1, 2, 2, 13, 6, 26, 4, 17, 1, 10, 2, 1, 2, 2, 1, 1

Offset

1,2

Comments

The 3x+1 function (A014682), denoted by T(x) in the literature, is defined as T(x) = (3x+1)/2 if x is odd, T(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  12  13  14  ...
  --------------------------------------------------------------------
   1 |  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   2 |  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2, ...
   3 |  3,  5,  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2, ...
   4 |  4,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   5 |  5,  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   6 |  6,  3,  5,  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   7 |  7, 11, 17, 26, 13, 20, 10,  5,  8,  4,  2,  1,  2,  1,  2, ...
   8 |  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2, ...
   9 |  9, 14,  7, 11, 17, 26, 13, 20, 10,  5,  8,  4,  2,  1,  2, ...
  10 | 10,  5,  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2,  1,  2, ...
  11 | 11, 17, 26, 13, 20, 10,  5,  8,  4,  2,  1,  2,  1,  2,  1, ...
  12 | 12,  6,  3,  5,  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2, ...
  13 | 13, 20, 10,  5,  8,  4,  2,  1,  2,  1,  2,  1,  2,  1,  2, ...
  14 | 14,  7, 11, 17, 26, 13, 20, 10,  5,  8,  4,  2,  1,  2,  1, ...
  15 | 15, 23, 35, 53, 80, 40, 20, 10,  5,  8,  4,  2,  1,  2,  1, ...
  ...

Mathematica

A365484list[dmax_]:=With[{a=Array[NestList[If[OddQ[#], (3#+1)/2, #/2]&, dmax-#, #]&, dmax, 0]}, Array[Diagonal[a, #]&, dmax, 1-dmax]]; A365484list[20] (* Generates 20 antidiagonals *)

Crossrefs

Cf. A014682, A347270 (equivalent for the Collatz function), A365495 (parity), A365991, A368300 (main diagonal).

Sequence in context: A362749 A264010 A071435 * A335814 A119428 A241815

Adjacent sequences: A365481 A365482 A365483 * A365485 A365486 A365487

Keyword

nonn,easy,tabl

Author

Paolo Xausa, Sep 06 2023

Status

approved