Sequence A112088 Motivation and Example?

[Graeme McRae]

It took me quite some time to understand how Ranier Rosenthal is counting 
rounds, before I obtained A112088 as the number of rounds necessary to kill 
all but one of n players of the Josephus Game with every third man out.  I 
first tried using Hugo Pfoertner's idea where the final survivor counts the 
times he was passed by the executioner including the final rendezvous.  But 
that method gives an entirely different sequence (A005428: 
1,2,3,4,6,9,14,21,31,47,70,105,158,...).  Then I realized it's the last 
*victim* who counts the number of times he was passed, including the final 
unfortunate rendezvous.  Presumably, he gasps the number with his last 
breath as he falls to the ground.  The last victim counts either the same 
number or one larger than the survivor, depending on where he stood 
initially, relative to the survivor.  (In case you were worried, no actual 
people were harmed in any of my experiments; I used gerbils.  I glued their 
feet to the table so they wouldn't leave their spot before I killed them. 
The only hard part was getting them to say the number of times I passed them 
as they were dying.)


----- Original Message ----- 
From: "Graeme McRae" <g_m at mcraefamily.com>
To: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Friday, June 09, 2006 2:20 PM
Subject: Re: Sequence A112088 Motivation and Example?


> Perhaps programming this sequence in Excel would be useful to you.  To do 
> that, perform the following steps:
> In cells A1, B1, and C1, place the values 2, 0, and 4.
> In cell D1, which will be the value of the sequence, place the formula 
> =B1+A1
> In cell A2, place the formula =IF(C1-D1<0,A1*2,A1)
> In cell B2, place the formula =FLOOR((2*A2-C2+1)/2,1)
> In cell C2, place the formula =IF(C1-D1<0,C1-D1+2*A2,C1-D1)
> Copy cell D1 to D2.
> Copy row 2 to rows 3 through 40.
> You will see the sequence in column D, if you have followed these 
> instructions (I really think they're correct!).
> Hopefully, you will see some similarities between the numbers in these 
> cells and the numbers that come up in your delightful game of executing 
> people in a circle.
>
>
> ----- Original Message ----- 
> From: "Rainer Rosenthal" <r.rosenthal at web.de>
> To: <neoneye at gmail.com>; <seqfan at ext.jussieu.fr>
> Cc: <fb at frank-buss.de>
> Sent: Friday, June 09, 2006 1:25 PM
> Subject: Sequence A112088 Motivation and Example?
>
>
>> To SeqFan and to author Simon Strandgaard:
>>
>> I am very much interested in this sequence:
>> http://www.research.att.com/~njas/sequences/A112088.
>>
>> I don't understand how Simon generated this
>> sequence...
>
>