A350279 Irregular triangle T(n,k) read by rows in which row n lists the iterates of the Farkas map (A349407) from 2*n - 1 to 1.

1, 3, 1, 5, 3, 1, 7, 11, 17, 9, 3, 1, 9, 3, 1, 11, 17, 9, 3, 1, 13, 7, 11, 17, 9, 3, 1, 15, 5, 3, 1, 17, 9, 3, 1, 19, 29, 15, 5, 3, 1, 21, 7, 11, 17, 9, 3, 1, 23, 35, 53, 27, 9, 3, 1, 25, 13, 7, 11, 17, 9, 3, 1, 27, 9, 3, 1, 29, 15, 5, 3, 1

Offset: 1

Author

Paolo Xausa, Dec 22 2021

Keywords

• nonn
• easy
• tabf

Examples

			Written as an irregular triangle, the sequence begins:
  n\k   1   2   3   4   5   6   7
  -------------------------------
   1:   1
   2:   3   1
   3:   5   3   1
   4:   7  11  17   9   3   1
   5:   9   3   1
   6:  11  17   9   3   1
   7:  13   7  11  17   9   3   1
   8:  15   5   3   1
   9:  17   9   3   1
  10:  19  29  15   5   3   1
  11:  21   7  11  17   9   3   1
  12:  23  35  53  27   9   3   1
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12301 (rows 1..1000 of triangle, flattened).
• H. M. Farkas, "Variants of the 3N+1 Conjecture and Multiplicative Semigroups", in Entov, Pinchover and Sageev, Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics, vol. 387, American Mathematical Society, 2005, p. 121.
• J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, p. 74.

Crossrefs

Cf. A349407, A375909 (# of iterations), A375910 (row sums), A375911 (row maxs).

Cf. A070165.

Programs

Mathematica

FarkasStep[x_] := Which[Divisible[x, 3], x/3, Mod[x, 4] == 3, (3*x + 1)/2, True, (x + 1)/2];
Array[Most[FixedPointList[FarkasStep, 2*# - 1]] &, 15] (* Paolo Xausa, Sep 03 2024 *)

Formula

T(n,1) = 2*n-1; T(n,k) = A349407((T(n,k-1)+1)/2), where n >= 1 and k >= 2.