A349407 The Farkas map: a(n) = x/3 if x mod 3 = 0; a(n) = (3x+1)/2 if x mod 3 <> 0 and x mod 4 = 3; a(n) = (x+1)/2 if x mod 3 <> 0 and x mod 4 = 1, where x = 2n-1.

1, 1, 3, 11, 3, 17, 7, 5, 9, 29, 7, 35, 13, 9, 15, 47, 11, 53, 19, 13, 21, 65, 15, 71, 25, 17, 27, 83, 19, 89, 31, 21, 33, 101, 23, 107, 37, 25, 39, 119, 27, 125, 43, 29, 45, 137, 31, 143, 49, 33, 51, 155, 35, 161, 55, 37, 57, 173, 39, 179, 61, 41, 63, 191, 43

Offset

1,3

Comments

The official Farkas map is given by: if x mod 3 = 0, F(x) = x/3; if x mod 4 = 3, F(x) = (3x+1)/2; otherwise F(x) = (x+1)/2. The map takes a positive odd integer and produces a positive odd integer.

Farkas proves that the trajectory of the iterates of the map starting from any positive odd integer always reaches 1.

If displayed as a rectangular array with six columns, the columns include A016921, A016813, A016945, A004767, A239129 (see example). - Omar E. Pol, Jan 01 2022

References

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.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

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

Example

From Omar E. Pol, Jan 01 2022: (Start)
Written as a rectangular array with six columns read by rows the sequence begins:
   1,  1,  3,  11,  3,  17;
   7,  5,  9,  29,  7,  35;
  13,  9, 15,  47, 11,  53;
  19, 13, 21,  65, 15,  71;
  25, 17, 27,  83, 19,  89;
  31, 21, 33, 101, 23, 107;
  37, 25, 39, 119, 27, 125;
  43, 29, 45, 137, 31, 143;
  49, 33, 51, 155, 35, 161;
  55, 37, 57, 173, 39, 179;
...
(End)

Mathematica

nterms=100; Table[x=2n-1; If[Mod[x, 3]==0, x/=3, If[Mod[x, 4]==3, x=(3x+1)/2, x=(x+1)/2]]; x, {n, nterms}]
(* Second program *)
nterms=100; LinearRecurrence[{0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1}, {1, 1, 3, 11, 3, 17, 7, 5, 9, 29, 7, 35}, nterms]
Table[Which[Mod[n, 3]==0, n/3, Mod[n, 4]==3, (3n+1)/2, True, (n+1)/2], {n, 1, 200, 2}] (* Harvey P. Dale, May 15 2022 *)

Prog

(PARI) a(n)=if (n%3==2, 2*n\3, if (n%2, n, 3*n-1)) \\ Charles R Greathouse IV, Nov 16 2021
(Python)
def a(n):
    x = 2*n - 1
    return x//3 if x%3 == 0 else ((3*x+1)//2 if x%4 == 3 else (x+1)//2)
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Nov 16 2021

Crossrefs

Cf. A006370, A014682, A016813, A016921, A016945, A004767, A239129, A350279, A350515.

Sequence in context: A359990 A170856 A176781 * A306367 A226625 A210610

Adjacent sequences: A349404 A349405 A349406 * A349408 A349409 A349410

Keyword

nonn,easy

Author

Paolo Xausa, Nov 16 2021

Status

approved