cp's OEIS Frontend

A075884 Image of n at the second step of the 3x+1 algorithm.

%I A075884 #35 Feb 16 2025 08:32:47
%S A075884 2,4,5,1,8,10,11,2,14,16,17,3,20,22,23,4,26,28,29,5,32,34,35,6,38,40,
%T A075884 41,7,44,46,47,8,50,52,53,9,56,58,59,10,62,64,65,11,68,70,71,12,74,76,
%U A075884 77,13,80,82,83,14,86,88,89,15,92,94,95,16,98,100,101,17,104,106,107
%N A075884 Image of n at the second step of the 3x+1 algorithm.
%C A075884 Also known as the Collatz Problem, Syracuse Algorithm or Hailstone Problem. Let syr(m,n) be the image of n at the m-th step. for m=2, k>=0 we get: syr(2,4k)=k, syr(2,4k+1)=6k+2, syr(2,4k+2)=6k+4, syr(2,4k+3)=6k+5.
%D A075884 David Wells, Penguin Dictionary of Curious and Interesting Numbers
%H A075884 Harvey P. Dale, <a href="/A075884/b075884.txt">Table of n, a(n) for n = 1..1000</a>
%H A075884 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A075884 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A075884 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,-1).
%F A075884 G.f.: x*(x^6 +2*x^5 +4*x^4 +x^3 +5*x^2 +4*x +2)/(1-x^4)^2.
%F A075884 a(n) = (6*n +(55*n+4)*m -6*(5*n-2)*m^2 +(5*n-4)*m^3)/24, m=(n mod 4). - _Zak Seidov_, Sep 14 2006
%F A075884 From _Federico Provvedi_, Oct 17 2021: (Start)
%F A075884 Dirichlet g.f.: ((3*4^s - 10)*zeta(s-1) + (4^s + 2^s - 2)*zeta(s))/2^(2s+1).
%F A075884 a(n) = (n*(19-5*i^(2*n)) - (5*n+4)*(i^n + (-i)^n) + 8)/16, where i*i = -1. (End)
%F A075884 a(n) = 2*a(n-4) - a(n-8). - _Wesley Ivan Hurt_, Apr 16 2023
%e A075884 1->4->2, 2->1->4, 3->10->5, 4->2->1, ...
%t A075884 Table[Nest[If[EvenQ[#],#/2,3#+1]&,n,2],{n,80}] (* _Harvey P. Dale_, Nov 15 2012 *)
%o A075884 (PARI) x='x+O('x^80); Vec(x*(x^6+2*x^5+4*x^4+x^3+5*x^2+4*x+2)/(1-x^4)^2) \\ _G. C. Greubel_, Oct 16 2018
%o A075884 (Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(x^6+2*x^5+4*x^4+x^3+5*x^2+4*x+2)/(1-x^4)^2));
%Y A075884 Cf. A006370 (the sequence at step 1), A076536 (at step 3).
%Y A075884 Column k=2 of A347270.
%K A075884 easy,nonn
%O A075884 1,1
%A A075884 Bruce Corrigan (scentman(AT)myfamily.com), Oct 16 2002