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

1,2

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.

Formula

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

Example

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

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 *)

Crossrefs

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

Cf. A070165.

Sequence in context: A159291 A122510 A102662 * A142048 A117563 A060439

Adjacent sequences: A350276 A350277 A350278 * A350280 A350281 A350282

Keyword

nonn,easy,tabf

Author

Paolo Xausa, Dec 22 2021

Status

approved