login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A327565 Number of transfers of marbles between two sets until the first repetition. 2
2, 3, 4, 3, 5, 4, 4, 5, 6, 4, 5, 6, 5, 5, 6, 5, 7, 6, 5, 7, 6, 5, 7, 6, 6, 7, 6, 6, 7, 6, 6, 7, 8, 6, 7, 8, 6, 7, 8, 6, 7, 8, 6, 7, 8, 6, 7, 8, 7, 7, 8, 7, 7, 8, 7, 7, 8, 7, 7, 8, 7, 7, 8, 7, 9, 8, 7, 9, 8, 7, 9, 8, 7, 9, 8, 7, 9, 8, 7, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There are initially n marbles in both sets. In the first turn, half of the marbles of set A are transferred to set B, rounding to the upper integer when halving. In the second turn, half of the marbles of set B are transferred back to set A, following the same rule. The game goes on back and forth until we reach a distribution already encountered.

a(n) is then the number of steps until the first repetition occurs.

First occurrence of a(n) = m > 1 in this sequence: 1, 2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049...

Conjecture: for m > 2, the first occurrence of a(n) = m is for n = 2^(m-3) + 1.

LINKS

Table of n, a(n) for n=1..80.

FORMULA

For m > 1, first occurrence of a(n) = m is for n = A094373(m-1) (conjectured).

EXAMPLE

For n = 3, (SetA ; SetB):

(3 ; 3), ceiling(3/2)=2 marbles get transferred,

(1 ; 5), ceiling(5/2)=3 marbles get transferred,

(4 ; 2), ceiling(4/2)=2 marbles get transferred,

(2 ; 4), ceiling(4/2)=2 marbles get transferred,

(4 ; 2), this is a repetition, it took 4 steps to get there, so a(3) = 4.

For n = 9, (SetA ; SetB):

(9 ; 9), (4 ; 14), (11 ; 7), (5 ; 13), (12 ; 6), (6 ; 12), (12 ; 6) which is a repetition, so a(9) = 6.

PROG

(PARI) a(n)={my(v=vector(2*n+1), r=n, f=1, c=0); while(!v[1+r], v[1+r]=1; r=if(f, r-ceil(r/2), r+ceil((2*n-r)/2)); c++; f=!f); c} \\ Andrew Howroyd, Sep 17 2019

CROSSREFS

Cf. A094373; A327613 (three sets), A327614 (four sets).

Sequence in context: A305890 A305801 A075850 * A054437 A287821 A299757

Adjacent sequences:  A327562 A327563 A327564 * A327566 A327568 A327569

KEYWORD

nonn

AUTHOR

Tristan Cam, Sep 17 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 7 15:56 EDT 2020. Contains 333306 sequences. (Running on oeis4.)