cp's OEIS Frontend

A160967 Numbers of the form (4^k - 1)/3 or 2^k.

%I A160967 #29 Sep 08 2022 08:45:45
%S A160967 0,1,2,4,5,8,16,21,32,64,85,128,256,341,512,1024,1365,2048,4096,5461,
%T A160967 8192,16384,21845,32768,65536,87381,131072,262144,349525,524288,
%U A160967 1048576,1398101,2097152,4194304,5592405,8388608,16777216,22369621,33554432
%N A160967 Numbers of the form (4^k - 1)/3 or 2^k.
%C A160967 A139391(a(n)) = 1 for n > 0.
%C A160967 Except for the first seven terms, the 3x + 1 trajectory for every number in this sequence includes 32. - _Alonso del Arte_, Jan 01 2015
%C A160967 For n>0, the Collatz-function starting with a(n) will terminate at 1. This is because all numbers of the form 2^k will terminate at 1, and ((4^k - 1)/3)*3 + 1 = 4^k = 2^2k. - _Bob Selcoe_, Apr 03 2015
%H A160967 Reinhard Zumkeller, <a href="/A160967/b160967.txt">Table of n, a(n) for n = 0..1000</a>
%H A160967 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,5,0,0,-4).
%F A160967 G.f.: x*(1 + 2*x + 4*x^2 - 2*x^4 - 4*x^5)/(1-5*x^3+4*x^6).
%F A160967 a(n+6) = 5*a(n+3) - 4*a(n) for n >= 1. - _Robert Israel_, Jan 01 2015
%p A160967 0, seq(op([(4^j-1)/3, 4^j/2,4^j]),j=1..20); # _Robert Israel_, Jan 01 2015
%t A160967 Union[2^Range[0, 25], (4^Range[0, 13] - 1)/3] (* _Alonso del Arte_, Jan 01 2015 *)
%t A160967 Join[{0}, LinearRecurrence[{0, 0, 5, 0, 0, -4}, {1, 2, 4, 5, 8, 16}, 50]] (* _Vincenzo Librandi_, Apr 03 2015 *)
%o A160967 (Haskell)
%o A160967 a160967 n = a160967_list !! n
%o A160967 a160967_list = m a000079_list a002450_list where
%o A160967    m xs'@(x:xs) ys'@(y:ys) | x < y     = x : m xs ys'
%o A160967                            | x == y    = x : m xs ys
%o A160967                            | otherwise = y : m xs' ys
%o A160967 -- _Reinhard Zumkeller_, Oct 03 2012
%o A160967 (Magma) I:=[0,1,2,4,5,8,16]; [n le 7 select I[n] else 5*Self(n-3)-4*Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Apr 03 2015
%Y A160967 Union of A002450 and A000079.
%K A160967 nonn,easy
%O A160967 0,3
%A A160967 _Reinhard Zumkeller_, May 31 2009