OFFSET
0,4
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.
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).
FORMULA
a(n) = (A349407(n+1)-1)/2.
a(n) = 2*a(n-6)-a(n-12). - Wesley Ivan Hurt, Jan 03 2022
EXAMPLE
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)
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]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paolo Xausa, Jan 02 2022
STATUS
approved