A033138 -id:A033138 - cp's OEIS frontend

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, 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

Offset: 1

Ryohei Miyadera, Jun 04 2006

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}.
Here we let p = 5 to produce the above sequence, but p can be an arbitrary positive integer. By letting p = 2, 3, 4, 6, 7 we can produce sequences A000975, A033138, A083593, A195904 and A117302, respectively.
The number of cases for each of the situations identified below by (0), (1), ..., (t), where t = floor((n-m)/p), can be calculated separately:
(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}. There are binomial(n-1,m-1) cases for this situation.
(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}. There are binomial(n-p-1,m-1) cases for this situation.
...
(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}. There are binomial(n-p*t-1,m-1) cases for this situation.
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).

			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.

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. Miyadera, General Theory of Russian Roulette, Mathematica source.
R. Miyadera, Interesting patterns of fractions, Archimedes' Laboratory.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,1,-2).

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

Partial sums of A349842.

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

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

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}] (* end of program *)
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 *)

for(n=1,50, print1(floor(2^(n+4)/31), ", ")) \\ G. C. Greubel, Oct 11 2017

a(n) = floor(2^(n+4)/31), which is obtained by letting p=5 in a_p(n) = (2^(n + p-1) - 2^((n-1) mod p))/(2^p - 1).
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)

A155803 A023001 interleaved with 2A023001 and 4A023001.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 9, 18, 36, 73, 146, 292, 585, 1170, 2340, 4681, 9362, 18724, 37449, 74898, 149796, 299593, 599186, 1198372, 2396745, 4793490, 9586980, 19173961, 38347922, 76695844, 153391689, 306783378, 613566756, 1227133513, 2454267026, 4908534052

Offset: 0

Paul Curtz, Jan 27 2009

A033138 with three zeros prepended. - Joerg Arndt, Mar 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

[Floor(2^n/7): n in [0..40]]; // Vincenzo Librandi, Sep 17 2011

A023001 := proc(n) (8^n-1)/7; end: A155803 := proc(n) RETURN( A023001(n),2*A023001(n),4*A023001(n)) ; end: L := [seq(A155803(n),n=0..30)] ; # R. J. Mathar, Jul 23 2009
seq(floor(2^n/7),n=0..30) # Mircea Merca, Dec 22 2010

CoefficientList[Series[x^3/(1-2 x-x^3+2 x^4), {x,0,50}],x]  (* Harvey P. Dale, Mar 13 2011 *)

a(3n) = A023001(n). a(3n+1) = 2*A023001(n) = A125835(n). a(3n+2) = 4*A023001(n).
a(n) = a(n-3)+2^(n-3) = a(n-3)+A000079(n-3). Here, a(.) can also be one of its higher order differences.
a(n) = 2*a(n-1)+a(n-3)-2*a(n-4). G.f.: x^3/((x-1)*(2*x-1)*(1+x+x^2)). [R. J. Mathar, Jul 23 2009]
a(n) = floor(2^n/7). [Mircea Merca, Dec 22 2010]

Edited and extended by R. J. Mathar, Jul 23 2009

A052935 Expansion of (2-2x-x^3)/((1-2x)*(1-x^3)).

Original entry on oeis.org

2, 2, 4, 9, 16, 32, 65, 128, 256, 513, 1024, 2048, 4097, 8192, 16384, 32769, 65536, 131072, 262145, 524288, 1048576, 2097153, 4194304, 8388608, 16777217, 33554432, 67108864, 134217729, 268435456, 536870912, 1073741825, 2147483648

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 923
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

a:=[2,2,4,9];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-3]-2*a[n-4]; od; a; # G. C. Greubel, Oct 18 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-x^3)/((1-2*x)*(1-x^3)) )); // G. C. Greubel, Oct 18 2019

spec:= [S,{S=Union(Sequence(Prod(Z,Z,Z)),Sequence(Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((2-2*x-x^3)/((1-2*x)*(1-x^3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 18 2019

CoefficientList[Series[(2-2*x-x^3)/((1-2*x)*(1-x^3)), {x, 0, 40}], x] (* G. C. Greubel, Oct 05 2017 *)
LinearRecurrence[{2,0,1,-2}, {2,2,4,9}, 40] (* G. C. Greubel, Oct 18 2019 *)

my(x='x+O('x^40)); Vec((2-2*x-x^3)/((1-2*x)*(1-x^3))) \\ G. C. Greubel, Oct 05 2017

def A052935_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((2-2*x-x^3)/((1-2*x)*(1-x^3))).list()
A052935_list(40) # G. C. Greubel, Oct 18 2019

G.f.: (2-2*x-x^3)/((1-x^3)*(1-2*x)).
a(n) = a(n-1) + a(n-2) + 2*a(n-3) - 1.
a(n) = 2^n + Sum_{alpha=RootOf(-1+z^3)} alpha^(-n)/3.
a(n) = 2*A033138(n+1) - 2*A033138(n) - A033138(n-2). - R. J. Mathar, Nov 28 2011

More terms from James Sellers, Jun 05 2000

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A155803 A023001 interleaved with 2*A023001 and 4*A023001.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

Formula

Extensions

A052935 Expansion of (2-2*x-x^3)/((1-2*x)*(1-x^3)).

Views

Author

Keywords

Links

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A155803 A023001 interleaved with 2A023001 and 4A023001.

A052935 Expansion of (2-2x-x^3)/((1-2x)*(1-x^3)).