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
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
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.
Comments