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A168221 a(n) = A006368(A006368(n)).

%I A168221 #50 Feb 07 2024 16:03:08
%S A168221 0,1,2,3,9,6,7,4,18,5,11,12,27,15,16,8,36,10,20,21,45,24,25,13,54,14,
%T A168221 29,30,63,33,34,17,72,19,38,39,81,42,43,22,90,23,47,48,99,51,52,26,
%U A168221 108,28,56,57,117,60,61,31,126,32,65,66,135,69,70,35,144,37,74,75,153,78,79,40
%N A168221 a(n) = A006368(A006368(n)).
%C A168221 Inverse integer permutation to A168222;
%C A168221 a(A006369(n)) = A006368(n).
%H A168221 A.H.M. Smeets, <a href="/A168221/b168221.txt">Table of n, a(n) for n = 0..20000</a>
%H A168221 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A168221 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A168221 <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1).
%F A168221 From _Luce ETIENNE_, Aug 14 2019: (Start)
%F A168221 a(n) = 2*a(n-16) - a(n-32).
%F A168221 a(n) = (-18*(40*m^7 - 973*m^6 + 9352*m^5 - 45115*m^4 + 114520*m^3 - 145432*m^2 + 75168*m - 10080)*floor(n/8) - m*(332*m^6 - 7973*m^5 + 75236*m^4 - 352835*m^3 + 855008*m^2 - 999992*m + 422664) + m*(4*m^6 - 105*m^5 + 1120*m^4 - 6195*m^3 + 18676*m^2 - 28980*m + 18000)*(-1)^(n/8))/10080 where m = n mod 8.
%F A168221 (End)
%F A168221 From _A.H.M. Smeets_, Aug 14 2019: (Start)
%F A168221 a(4*n) = 9*n.
%F A168221 a(8*n+1) = a(8*n-1)+1, n > 0.
%F A168221 a(8*n+3) = a(8*n+2)+1.
%F A168221 a(8*n+5) = a(8*n+3)+3 = a(8*n+2)+4.
%F A168221 a(8*n+6) = a(8*n+5)+1 = a(8*n+3)+4 = a(8*n+2)+5.
%F A168221 a(16*n+1) = 9*n+1.
%F A168221 a(16*n+2) = 18*n+2.
%F A168221 a(16*n+3) = a(16*n+2)+1 = 18*n+3.
%F A168221 a(16*n+5) = a(16*n+3)+3 = 18*n+6.
%F A168221 a(16*n+6) = a(16*n+5)+1 = 18*n+7.
%F A168221 a(16*n+7) = (a(16*n+6)+1)/2 = 9*n+4.
%F A168221 a(16*n+9) = 9*n+5.
%F A168221 a(16*n+10) = 2*a(16*n+9)+1 = 18*n+11.
%F A168221 a(16*n+11) = a(16*n+10)+1 = 18*n+12.
%F A168221 a(16*n+13) = a(16*n+11)+3 = 18*n+15.
%F A168221 a(16*n+14) = a(16*n+13) = 18*n+16.
%F A168221 a(16*n+15) = a(16*n+14)/2 = 9*n+8.
%F A168221 From this, (9*n-7)/16 <= a(n) <= 9*n/4.
%F A168221 (End)
%F A168221 From _Colin Barker_, Aug 23 2019: (Start)
%F A168221 G.f.: x*(1 - x + x^2)*(1 + 3*x + 5*x^2 + 11*x^3 + 12*x^4 + 8*x^5 + 10*x^7 + 14*x^8 + 13*x^9 + 8*x^10 + 13*x^11 + 14*x^12 + 10*x^13 + 8*x^15 + 12*x^16 + 11*x^17 + 5*x^18 + 3*x^19 + x^20) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2*(1 + x^4)^2*(1 + x^8)).
%F A168221 a(n) = a(n-8) + a(n-16) - a(n-24) for n>23.
%F A168221 (End)
%t A168221 Table[Nest[If[OddQ[#],Floor[(3#+2)/4],3#/2]&,n,2],{n,0,100}] (* _Paolo Xausa_, Dec 15 2023 *)
%t A168221 LinearRecurrence[{0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1},{0,1,2,3,9,6,7,4,18,5,11,12,27,15,16,8,36,10,20,21,45,24,25,13},80] (* _Harvey P. Dale_, Feb 07 2024 *)
%o A168221 (Python)
%o A168221 def A006368(n):
%o A168221     if n%2 == 0:
%o A168221         return 3*(n//2)
%o A168221     elif n%4 == 1:
%o A168221         return 3*(n//4)+1
%o A168221     else:
%o A168221         return 3*(n+1)//4-1
%o A168221 n = 0
%o A168221 while n < 30:
%o A168221     print(n,A006368(A006368(n)))
%o A168221     n = n+1 # _A.H.M. Smeets_, Aug 14 2019
%Y A168221 Cf. A006368, A006369, A168222.
%K A168221 nonn,easy
%O A168221 0,3
%A A168221 _Reinhard Zumkeller_, Nov 20 2009