A350515 - OEIS

%I #18 Jan 04 2022 05:25:48

%S 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,

%T 44,15,10,16,50,11,53,18,12,19,59,13,62,21,14,22,68,15,71,24,16,25,77,

%U 17,80,27,18,28,86,19,89,30,20,31,95,21,98,33,22,34,104

%N 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.

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

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

%C 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

%H H. M. Farkas, "Variants of the 3N+1 Conjecture and Multiplicative Semigroups", in Entov, Pinchover and Sageev, <a href="https://bookstore.ams.org/conm-387">Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics, vol. 387</a>, American Mathematical Society, 2005, p. 121.

%H Emre Yolcu, Scott Aaronson and Marijn J. H. Heule, <a href="https://arxiv.org/abs/2105.14697">An Automated Approach to the Collatz Conjecture</a>, arXiv:2105.14697 [cs.LO], 2021, pp. 21-25.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

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

%F a(n) = 2*a(n-6)-a(n-12). - _Wesley Ivan Hurt_, Jan 03 2022

%e From _Omar E. Pol_, Jan 02 2022: (Start)

%e Written as a rectangular array with six columns read by rows the sequence begins:

%e    0,  0,  1,  5,  1,  8;

%e    3,  2,  4, 14,  3, 17;

%e    6,  4,  7, 23,  5, 26;

%e    9,  6, 10, 32,  7, 35;

%e   12,  8, 13, 41,  9, 44;

%e   15, 10, 16, 50, 11, 53;

%e   18, 12, 19, 59, 13, 62;

%e   21, 14, 22, 68, 15, 71;

%e   24, 16, 25, 77, 17, 80;

%e   27, 18, 28, 86, 19, 89;

%e   30, 20, 31, 95, 21, 98;

%e ...

%e (End)

%t 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}]

%t (* Second program *)

%t 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]

%o (Python)

%o def a(n):

%o     r = n%6

%o     if r == 1 or r == 4: return (n-1)//3

%o     if r == 0 or r == 2: return n//2

%o     if r == 3 or r == 5: return (3*n+1)//2

%o print([a(n) for n in range(70)]) # _Michael S. Branicky_, Jan 02 2022

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

%K nonn,easy

%O 0,4

%A _Paolo Xausa_, Jan 02 2022