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

1,1

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).

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A006368.

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

Sequence in context: A353538 A220099 A067737 * A111494 A092025 A250329

Adjacent sequences: A217215 A217216 A217217 * A217219 A217220 A217221

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 04 2012

Status

approved