A217218 Trajectory of 44 under the map k -> A006368(k).

44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59

Offset: 1

Author

N. J. A. Sloane, Oct 04 2012

Keywords

• nonn
• easy

Comments

Periodic with period length 12.

It is believed that this is the longest trajectory that cycles (the others are {1}, {2,3}, {4,6,9,7,5}).

References

• See also references and links in A006368.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.
• John L Simons, Cycles and divergent trajectories for a class of permutation sequences, arXiv:2205.10582 [math.NT], 2022.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A006368.

Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

Programs

Haskell

a217218 n = a217218_list !! (n-1)
a217218_list = iterate a006368 44  -- Reinhard Zumkeller, Apr 06 2013

Magma

&cat[ [44,66,99,74,111,83,62,93,70,105,79,59]: n in [0..9] ]; // Vincenzo Librandi, Jun 28 2015

Mathematica

t={44}; While[n=t[[-1]]; s=If[EvenQ[n], 3*n/2, Round[3*n/4]]; Length[t]<100&&!MemberQ[t, s], AppendTo[t, s]]; t (* Vincenzo Librandi, Jun 28 2015 *)
PadRight[{},120,{44,66,99,74,111,83,62,93,70,105,79,59}] (* Harvey P. Dale, Jul 30 2025 *)

PARI

Vec(x*(44 + 66*x + 99*x^2 + 74*x^3 + 111*x^4 + 83*x^5 + 62*x^6 + 93*x^7 + 70*x^8 + 105*x^9 + 79*x^10 + 59*x^11) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)) + O(x^40)) \\ Colin Barker, Aug 16 2019

Formula

a(n+1) = A006368(a(n)).

From Colin Barker, Aug 16 2019: (Start)

G.f.: x*(44 + 66*x + 99*x^2 + 74*x^3 + 111*x^4 + 83*x^5 + 62*x^6 + 93*x^7 + 70*x^8 + 105*x^9 + 79*x^10 + 59*x^11) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)).

a(n) = a(n-12) for n>12.

(End)