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A119610 Number of cases in which the first player is killed in a Russian roulette game where 5 players use a gun with n chambers and the number of bullets can be from 1 to n. Players do not rotate the cylinder after the game starts. 1
1, 2, 4, 8, 16, 33, 66, 132, 264, 528, 1057, 2114, 4228, 8456, 16912, 33825, 67650, 135300, 270600, 541200, 1082401, 2164802, 4329604, 8659208, 17318416, 34636833, 69273666, 138547332, 277094664, 554189328, 1108378657, 2216757314 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Here we let p = 5 to produce the above sequence, but p can be an arbitrary natural number. By letting p = 2, 3, 4, 6, 7 we can produce sequences A000975, A033138, A083593, A195904 and A117302. We denote by U(p,n,m) the number of the cases in which the first player is killed in a Russian roulette game where p players use a gun with n chambers and m bullets. They never rotate the cylinder after the game starts. The chambers can be represented by the list {1,2,...,n}.

We calculate the number of cases for each of the situations identified below by (0), (1), ..., (t) separately, where t = floor((n-m)/p):

(0) The first player is killed when one bullet is in the first chamber and the remaining (m-1) bullets are in chambers {2,3,...,n}. We have binomial(n-1,m-1) cases for this.

(1) The first player is killed when one bullet is in the (p+1)-th chamber and the rest of the bullets are in chambers {p+2,..,n}. We have binomial(n-p-1,m-1) cases for this.

...

(t) The first player is killed when one bullet is in the (p*t+1)-th chamber and the remaining bullets are in chambers {p*t+2,...,n}. We have binomial(n-p*t-1,m-1) cases for this.

Therefore U(p,n,m) = Sum_{z=0..t} binomial(n-p*z-1,m-1), where t = floor((n-m)/p). Let A(p,n) be the number of the cases in which the first player is killed when p players use a gun with n chambers and the number of bullets can be from 1 to n. Then A(p,n) = Sum_{m=1..n} U(p,n,m).

REFERENCES

R. Miyadera, "General Theory of Russian Roulette." Mathematica source.

R. Miyadera, Mathematical Theory of Magic Fruits Archimedes-lab.

LINKS

Table of n, a(n) for n=1..32.

R. Miyadera, General Theory of Russian Roulette, Mathematica source.

R. Miyadera, Interesting patterns of fractions, Archimedes-lab

Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,1,-2).

FORMULA

Let p = 5 and a(n) = (2^(n + p-1) - 2^((n-1) mod p))/(2^p - 1). In the following we present the formula based on the theory we used to define our sequence as a Mathematica code. The above formula is much easier to use, but the Mathematica code has an important mathematical meaning in it.

a(n) = floor(2^(n+4)/31), which is obtained by letting p=5 in the formula above.

From Joerg Arndt, Jan 08 2011: (Start)

G.f.: x / ( (x-1)*(2*x-1)*(x^4+x^3+x^2+x+1) ).

a(n) = +2*a(n-1) +a(n-5) -2*a(n-6). (End)

EXAMPLE

If the number of chambers is 3, then the number of the bullets can be 1, 2, or 3. The first player is killed when one bullet is in the first chamber, and the remaining bullets are in the second and third chambers. The only cases are {{1, 0, 0}, {1, 1, 0}, {1, 0, 1}, {1, 1, 1}}, where we denote by 1 a chamber that contains a bullet. Therefore a(3) = 4.

MAPLE

seq(floor(2^(n+4)/31), n = 1..32); # Mircea Merca, Dec 22 2010

MATHEMATICA

U[p_, n_, m_, v_]:=Block[{t}, t=Floor[(1+p-m+n-v)/p]; Sum[Binomial[n-v-p*z, m-1], {z, 0, t-1}]]; A[p_, n_, v_]:=Sum[U[p, n, k, v], {k, 1, n}]; (* Here we let p = 5 to produce the above sequence, but this code can produce A000975, A033138, A083593, A195904, A117302 for p = 2, 3, 4, 6, 7. *) Table[B[5, n, 1], {n, 1, 20}]

CoefficientList[ Series[ 1/(2x^6 - x^5 - 2x + 1), {x, 0, 32}], x] (* or *)

LinearRecurrence[{2, 0, 0, 0, 1, -2}, {1, 2, 4, 8, 16, 33}, 32] (* Robert G. Wilson v, Mar 12 2015 *)

PROG

(MAGMA) I:=[1, 2, 4, 8, 16, 33]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-5)-2*Self(n-6): n in [1..40]]; // Vincenzo Librandi, Mar 18 2015

CROSSREFS

Cf. A000975, A033138, A083593, A195904, A117302.

Sequence in context: A137181 A272698 A036373 * A121485 A182442 A098588

Adjacent sequences:  A119607 A119608 A119609 * A119611 A119612 A119613

KEYWORD

easy,nonn,uned

AUTHOR

Ryohei Miyadera, Jun 04 2006

STATUS

approved

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Last modified March 25 21:19 EDT 2017. Contains 284111 sequences.