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A101229 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Gives a particular sequence of hailstone numbers which, perhaps, may be considered as a central axis for all the hailstone numbers 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). [From 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

Table of n, a(n) for n=1..39.

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. [From Jaroslav Krizek, Aug 17 2009]

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). - Colin Barker, Apr 28 2013

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: 4x2=8

CROSSREFS

Cf. A070165, A006577 A006667 A006666 A070167.

Sequence in context: A115314 A062039 A035492 * A057176 A194671 A131398

Adjacent sequences:  A101226 A101227 A101228 * A101230 A101231 A101232

KEYWORD

nonn,uned,easy

AUTHOR

Alexandre Wajnberg, Jan 22 2005

EXTENSIONS

More terms from Joshua Zucker, May 18 2006

STATUS

approved

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Last modified September 24 04:27 EDT 2017. Contains 292403 sequences.