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A076536 Image of n at the third step in the 3x+1 Problem: syr(3,n).

%I A076536 #31 Feb 16 2025 08:32:47
%S A076536 1,2,16,4,4,5,34,1,7,8,52,10,10,11,70,2,13,14,88,16,16,17,106,3,19,20,
%T A076536 124,22,22,23,142,4,25,26,160,28,28,29,178,5,31,32,196,34,34,35,214,6,
%U A076536 37,38,232,40,40,41,250,7,43,44,268,46,46,47,286,8,49,50,304,52,52,53
%N A076536 Image of n at the third step in the 3x+1 Problem: syr(3,n).
%C A076536 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=3, k>=0 we get: syr(3,8k)=k, syr(3,8k+1)=6k+1, syr(3,8k+2)=6k+2, syr(3,8k+3)=36k+16, syr(3,8k+4)=6k+4, syr(3,8k+5)=6k+4, syr(3,8k+6)=6k+5, syr(3,8k+7)=36k+34.
%D A076536 David Wells, Penguin Dictionary of Curious and Interesting Numbers.
%H A076536 G. C. Greubel, <a href="/A076536/b076536.txt">Table of n, a(n) for n = 1..10000</a>
%H A076536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A076536 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A076536 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).
%F A076536 G.f.: x*(1 + 2*x + 16*x^2 + 4*x^3 + 4*x^4 + 5*x^5 + 34*x^6 + x^7 + 5*x^8 + 4*x^9 + 20*x^10 + 2*x^11 + 2*x^12 + x^13 + 2*x^14)/(1 - x^8)^2.
%F A076536 a(n) = (1/64)*(103*n + 60 + i^n*(36*i - n*(5-60*i)) - i^(2n)*(65*n+28) - i^(3n)(36*i+n*(5+60*i)) - i^(n/2)*(8+5n)*(1 + i^n + i^(2n) + i^(3n))). - _Federico Provvedi_, Nov 23 2021
%e A076536 1->4->2->1; 2->1->4->2; 3->10->5->16; ...
%t A076536 Rest[CoefficientList[Series[(x +2x^2 +16x^3 +4x^4 +4x^5 +5x^6 +34x^7 + x^8 +5x^9 +4x^10 +20x^11 +2x^12 +2x^13 +x^14 +2x^15)/(1-x^8)^2, {x, 0, 80}], x]] (* _G. C. Greubel_, Oct 16 2018 *)
%o A076536 (PARI) x='x+O('x^80); Vec(x*(1 +2*x +16*x^2 +4*x^3 +4*x^4 +5*x^5 +34*x^6 +x^7 +5*x^8 +4*x^9 +20*x^10 +2*x^11 +2*x^12 +x^13 +2*x^14)/(1-x^8)^2) \\ _G. C. Greubel_, Oct 16 2018
%o A076536 (Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 +2*x +16*x^2 +4*x^3 +4*x^4 +5*x^5 +34*x^6 +x^7 +5*x^8 +4*x^9 +20*x^10 +2*x^11 +2*x^12 +x^13 +2*x^14)/(1-x^8)^2)); // _G. C. Greubel_, Oct 16 2018
%Y A076536 Cf. A006370 (n at step 1), A075884 (n at step 2).
%Y A076536 Column k=3 of A347270.
%K A076536 easy,nonn
%O A076536 1,2
%A A076536 Bruce Corrigan (scentman(AT)myfamily.com), Oct 18 2002