A101229 Perfect inverse "3x+1 conjecture" (See comments for rules).

1, 2, 4, 1, 2, 4, 8, 16, 5, 10, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368

Offset

1,2

Comments

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.

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.

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

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Collatz Problem

Index entries for linear recurrences with constant coefficients, signature (2).

Formula

a(n) = 3*2^(n-11) = 2^(n-11) + 2^(n-10) for n >= 11. - Jaroslav Krizek, Aug 17 2009

From Colin Barker, Apr 28 2013: (Start)

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

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

Example

The first 4 is followed by 1 because 4 = 3*1 + 1, so rule 2: (4-1)/3 = 1;
the second 4 is followed by 8 because the 2nd rule has already been applied, so rule 1: 4*2 = 8.

Mathematica

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 *)
LinearRecurrence[{2}, {1, 2, 4, 1, 2, 4, 8, 16, 5, 10, 3}, 40] (* Harvey P. Dale, May 06 2023 *)

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A352866 A352883 A035492 * A057176 A194671 A331696

Adjacent sequences: A101226 A101227 A101228 * A101230 A101231 A101232

Keyword

nonn,easy

Author

Alexandre Wajnberg, Jan 22 2005

Extensions

More terms from Joshua Zucker, May 18 2006

Edited by G. C. Greubel, Mar 20 2019

Status

approved