A350515 a(n) = (n-1)/3 if n mod 3 = 1; a(n) = n/2 if n mod 6 = 0 or n mod 6 = 2; a(n) = (3n+1)/2 if n mod 6 = 3 or n mod 6 = 5.

0, 0, 1, 5, 1, 8, 3, 2, 4, 14, 3, 17, 6, 4, 7, 23, 5, 26, 9, 6, 10, 32, 7, 35, 12, 8, 13, 41, 9, 44, 15, 10, 16, 50, 11, 53, 18, 12, 19, 59, 13, 62, 21, 14, 22, 68, 15, 71, 24, 16, 25, 77, 17, 80, 27, 18, 28, 86, 19, 89, 30, 20, 31, 95, 21, 98, 33, 22, 34, 104

Offset: 0

Author

Paolo Xausa, Jan 02 2022

Keywords

• nonn
• easy

Comments

This is a variant of the Farkas map (A349407).

Yolcu, Aaronson and Heule prove that the trajectory of the iterates of the map starting from any nonnegative integer always reaches 0.

If displayed as a rectangular array with six columns, the columns are A008585, A005843, A016777, A017221, A005408, A017257 (see example). - Omar E. Pol, Jan 02 2022

Examples

			From _Omar E. Pol_, Jan 02 2022: (Start)
Written as a rectangular array with six columns read by rows the sequence begins:
   0,  0,  1,  5,  1,  8;
   3,  2,  4, 14,  3, 17;
   6,  4,  7, 23,  5, 26;
   9,  6, 10, 32,  7, 35;
  12,  8, 13, 41,  9, 44;
  15, 10, 16, 50, 11, 53;
  18, 12, 19, 59, 13, 62;
  21, 14, 22, 68, 15, 71;
  24, 16, 25, 77, 17, 80;
  27, 18, 28, 86, 19, 89;
  30, 20, 31, 95, 21, 98;
...
(End)
		

Links

• 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.
• Emre Yolcu, Scott Aaronson and Marijn J. H. Heule, An Automated Approach to the Collatz Conjecture, arXiv:2105.14697 [cs.LO], 2021, pp. 21-25.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Crossrefs

Cf. A005408, A005843, A006370, A014682, A016777, A017221, A017257, A349407, A350279.

Programs

Mathematica

nterms=100;Table[If[Mod[n,3]==1,(n-1)/3,If[Mod[n,6]==0||Mod[n,6]==2,n/2,(3n+1)/2]],{n,0,nterms-1}]
(* Second program *)
nterms=100;LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{0,0,1,5,1,8,3,2,4,14,3,17},nterms]

Python

def a(n):
    r = n%6
    if r == 1 or r == 4: return (n-1)//3
    if r == 0 or r == 2: return n//2
    if r == 3 or r == 5: return (3*n+1)//2
print([a(n) for n in range(70)]) # Michael S. Branicky, Jan 02 2022

Formula

a(n) = (A349407(n+1)-1)/2.

a(n) = 2*a(n-6)-a(n-12). - Wesley Ivan Hurt, Jan 03 2022