[seqfan] Re: the revolver sequence

[Max Alekseyev]

Thanks for explicit example - I've found a typo in my code.

Here is a formula and first 50 terms of your sequence:

{ a(n) = sum(z=0,n,sum(r=1,min(ceil(z-z^2/n)-1,n-z),
sumdiv(gcd([n,z,r]),d, eulerphi(d) * binomial(z/d - 1, r/d - 1) *
binomial((n-z)/d - 1, r/d - 1) )/r )) }

? vector(50,n,a(n))
%2 = [0, 0, 0, 0, 2, 3, 4, 5, 18, 43, 60, 84, 294, 472, 724, 1077,
3504, 5807, 10396, 15944, 43664, 84308, 137004, 217728, 582966,
1134304, 1883360, 3225812, 8964134, 15461472, 27796942, 45814765,
123307634, 233218013, 396304692, 663041846, 1754000726, 3300299482,
5679338110, 9942238090, 24989323012, 47334027696, 85892508384,
146496196464, 359597562016, 690754558978, 1256104705128,
2163505451106, 5289200995480, 10102216102758]

Regards,
Max

On Wed, Sep 23, 2009 at 10:25 PM, Tanya Khovanova
<mathoflove-seqfan at yahoo.com> wrote:
> This is my list for n=9:
>
> {{0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 0, 1, 1, 1}, {0, 0, 0,
>  0, 0, 1, 1, 1, 1}, {0, 0, 0, 0, 1, 0, 1, 1, 1}, {0, 0, 0, 0, 1, 1,
>  0, 1, 1}, {0, 0, 0, 0, 1, 1, 1, 0, 1}, {0, 0, 0, 1, 0, 0, 1, 1,
>  1}, {0, 0, 0, 1, 1, 0, 0, 1, 1}, {0, 0, 0, 1, 1, 1, 0, 0, 1}, {0, 0,
>   0, 0, 1, 1, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1, 1, 1}, {0, 0, 0, 1, 1,
>  0, 1, 1, 1}, {0, 0, 0, 1, 1, 1, 0, 1, 1}, {0, 0, 0, 1, 1, 1, 1, 0,
>  1}, {0, 0, 1, 0, 0, 1, 1, 1, 1}, {0, 0, 1, 1, 0, 0, 1, 1, 1}, {0, 0,
>   0, 1, 1, 1, 1, 1, 1}, {0, 0, 1, 1, 1, 1, 1, 1, 1}}
>
> --- On Wed, 9/23/09, Max Alekseyev <maxale at gmail.com> wrote:
>
>> From: Max Alekseyev <maxale at gmail.com>
>> Subject: [seqfan] Re: the revolver sequence
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Date: Wednesday, September 23, 2009, 6:58 PM
>> Hi Tanya,
>>
>> I've got different values starting with a(9) = 13, a(10) =
>> 32 etc.
>> Could you please double check your values or lists all 18
>> ways of
>> loading a revolver with n=9 chambers?
>>
>> Thanks,
>> Max
>>
>> On Wed, Sep 23, 2009 at 5:06 PM, Tanya Khovanova
>> <mathoflove-seqfan at yahoo.com>
>> wrote:
>> > Hello all,
>> >
>> > I just submitted a sequence related to the revolver
>> puzzle:
>> > http://blog.tanyakhovanova.com/?p=174
>> >
>> > I generalized it to revolvers having n chambers.
>> >
>> > I think there should be more sequences: for example,
>> the one when it is preferable to spin and the one when the
>> chances are the same.
>> >
>> > Also, my programs are usually not very efficient. It
>> will be nice if someone can add more terms to this
>> sequence.
>> >
>> > %I A165646
>> > %S A165646 0,0,0,0,2,3,4,5,18,43,60,84,294,472,724
>> > %N A165646 The revolver sequence.
>> > %C A165646 a(n) is in how many ways you can load a
>> revolver with n chambers, so that if a person survives after
>> the first shot, he/she has better chances of survival for
>> the second shot if the shooter continues rather than spins.
>> a(n) <= A000031(n) as A000031(n) counts the number of
>> possible revolver loadings.
>> > %H A165646 Tanya Khovanova, <a href="http://blog.tanyakhovanova.com/?p=174">Heard on the
>> Street</a>
>> > %e A165646 Based on the famous interview question
>> where the revolver has six chambers and the shooter loads
>> two adjacent bullets. As the answer to this question is to
>> continue, a(6) must be at least 1.
>> > %t A165646 << Combinatorica` colors[x_] :=
>> Table[PadLeft[Table[1, {n, i}], x], {i, 0, x}]
>> continueBetter[list_] := (len = Length[list]; c0 = 0; c00 =
>> 0; rotateN = 0; While[rotateN < len, newList =
>> RotateLeft[list, rotateN]; If[newList[[1]] == 0, c0++;
>> If[newList[[2]] == 0, c00++]]; rotateN++]; If[c0 > 0,
>> c00/c0 > c0/len, False]) continueNum[num_] := (neckNum =
>> Length[colors[num]]; ans = 0; count = 1; While[count <=
>> neckNum, ans = ans + Length[Select[ListNecklaces[num,
>> colors[num][[count]], Cyclic], continueBetter[#] &]];
>> count++]; ans) Table[continueNum[n], {n, 2, 15}]
>> > %Y A165646 A000031
>> > %K A165646 more,nonn
>> > %O A165646 1,5
>> > %A A165646 Tanya Khovanova (tanyakh(AT)yahoo.com), Sep
>> 23 2009
>> >
>> >
>> >
>> > _______________________________________________
>> >
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>> >
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