A101229 - OEIS

%I #27 Aug 30 2024 22:12:04

%S 1,2,4,1,2,4,8,16,5,10,3,6,12,24,48,96,192,384,768,1536,3072,6144,

%T 12288,24576,49152,98304,196608,393216,786432,1572864,3145728,6291456,

%U 12582912,25165824,50331648,100663296,201326592,402653184,805306368

%N Perfect inverse "3x+1 conjecture" (See comments for rules).

%C Perfect inverse "3x+1 conjecture": rule 1: multiply n by 2 to give n' = 2n. rule 2: when n'=(3x+1), do n"= (n'-1)/3 (n" integer) Additional rule: rule 2 is applied once for any number n' (otherwise, the sequence beginning with 1 would be the cycle "1 2 4 1 2 4 1 2 4 1..."); then apply rule 1.

%C This gives a particular sequence of hailstone numbers which may be considered as a central axis for all the hailstone number sequences. The perfect inverse "3x+1 conjecture" falls rapidly into the sequence 3 6 12 24 48 96... which will never give a number to which apply the 2nd rule.

%C a(n) for n >= 11 written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-11) times 0 (see A003953(n-10)). - _Jaroslav Krizek_, Aug 17 2009

%D R. K. Guy, Collatz's Sequence, Section E16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994.

%H G. C. Greubel, <a href="/A101229/b101229.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).

%F a(n) = 3*2^(n-11) = 2^(n-11) + 2^(n-10) for n >= 11. - _Jaroslav Krizek_, Aug 17 2009

%F From _Colin Barker_, Apr 28 2013: (Start)

%F a(n) = 2*a(n-1) for n>11.

%F G.f.: x*(17*x^10+27*x^8+7*x^3-1) / (2*x-1). (End)

%e The first 4 is followed by 1 because 4 = 3*1 + 1, so rule 2: (4-1)/3 = 1;

%e the second 4 is followed by 8 because the 2nd rule has already been applied, so rule 1: 4*2 = 8.

%t Rest[CoefficientList[Series[x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1), {x, 0, 45}], x]] (* _G. C. Greubel_, Mar 20 2019 *)

%t LinearRecurrence[{2},{1,2,4,1,2,4,8,16,5,10,3},40] (* _Harvey P. Dale_, May 06 2023 *)

%o (PARI) my(x='x+O('x^45)); Vec(x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1)) \\ _G. C. Greubel_, Mar 20 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 45); Coefficients(R!( x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1) )); // _G. C. Greubel_, Mar 20 2019

%o (Sage) a=(x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1)).series(x, 45).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Mar 20 2019

%Y Cf. A070165, A006577, A006667, A006666, A070167.

%K nonn,easy

%O 1,2

%A _Alexandre Wajnberg_, Jan 22 2005

%E More terms from _Joshua Zucker_, May 18 2006

%E Edited by _G. C. Greubel_, Mar 20 2019