This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A117302 #32 Sep 08 2022 08:45:24
%S A117302 1,2,4,8,16,32,64,129,258,516,1032,2064,4128,8256,16513,33026,66052,
%T A117302 132104,264208,528416,1056832,2113665,4227330,8454660,16909320,
%U A117302 33818640,67637280,135274560,270549121,541098242
%N A117302 Number of cases in which the first player gets killed in a Russian roulette game when 7 players use a gun with n chambers and the number of the bullets can be from 1 to n. In the game they do not rotate the cylinder after the game starts.
%C A117302 We denote by U[7,n,m] the number of cases that the first player gets killed in a Russian roulette game when 7 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 are going to calculate (0), (1), ..., (t) separately. (0) The first player gets killed when one bullet is in the first chamber and the remaining (m-1) bullets are in {2,3,...,n}. We have binomial(n-1,m-1) cases for this. (1) The first gets killed when one bullet is in the 8th chamber and the rest of the bullets are in {9,..,n}. We have binomial(n-8,m-1) cases for this. We continue to calculate and the last is (t), where t = floor((n-m)/7). (t) The first gets killed when one bullet is in (7t+1)-st chamber and the remaining bullets are in {7t+2,...,n}. We have binomial(n-7t-1,m-1) cases for this. Therefore U[7,n,m] = Sum_{z=0..t} binomial(n-7z-1,m-1), where t = floor((n-m)/7). Let A[7,n] be the number of cases in which the first player gets killed when 7 players use a gun with n chambers and the number of bullets can be from 1 to n. Then A[7,n] = Sum_{m=1..n} U[7,n,m].
%D A117302 Miyadera, R. "General Theory of Russian Roulette." Mathematica source.
%D A117302 Miyadera, R. Mathematical Theory of Magic Fruits Archimedes-lab.
%H A117302 G. C. Greubel, <a href="/A117302/b117302.txt">Table of n, a(n) for n = 1..1000</a>
%H A117302 R. Miyadera, <a href="http://library.wolfram.com/infocenter/MathSource/5710/">General Theory of Russian Roulette</a>, MathSource
%H A117302 R. Miyadera, Daisuke Minematsu, Satoshi Hashiba and Munetoshi Hashiba, <a href="http://www.archimedes-lab.org/fraction_patt/Magic_fruits.html">Theory of Magic Fruits</a>, Archimedes-lab, Interesting patterns of fractions
%H A117302 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,0,0,1,-2).
%F A117302 a(n) = (2^(n + 6) - 2^((n-1) mod 7))/(2^7 - 1).
%F A117302 a(n) = floor(2^(n+6)/127). - _Mircea Merca_, Dec 22 2010
%F A117302 From _Joerg Arndt_, Jan 08 2011: (Start)
%F A117302 G.f.: x/( (x-1)*(2*x-1)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) ).
%F A117302 a(n) = 2*a(n-1) + a(n-7) - 2*a(n-8). (End)
%e A117302 If the number of chambers is 3, then the number of the bullets can be 1,2,3. The first one get killed when one bullet is in the first chamber and the remaining bullets are in the second and the third chamber. All the cases are {{1, 0, 0}, {1, 1, 0}, {1, 0, 1}, {1, 1, 1}}, where we denote by 1 the chamber that contains the bullet. Therefore a(3) = 4.
%p A117302 A117302 := proc(n) floor(2^(n+6)/127) ; end proc:
%t A117302 U7[n_, m_]:= Block[{t}, t=Floor[(n-m)/7]; Sum[Binomial[n-1-7z, m-1], {z,0,t}]]; A7[n_]:= Sum[U7[n, m], {m,1,n}]; Table[A7[n], {n,1,40}]
%t A117302 LinearRecurrence[{2,0,0,0,0,0,1,-2}, {1,2,4,8,16,32,64,129}, 40] (* _G. C. Greubel_, May 07 2019 *)
%o A117302 (PARI) a(n)=2^(n+6)\127 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A117302 (Magma) [Floor(2^(n+6)/127): n in [1..40]]; // _G. C. Greubel_, May 07 2019
%o A117302 (Sage) [floor(2^(n+6)/127) for n in (1..40)] # _G. C. Greubel_, May 07 2019
%K A117302 nonn,easy
%O A117302 1,2
%A A117302 Tomohide Hashiba, Akihiro Hyogu, Hiroshi Matsui, _Ryohei Miyadera_, Yuta Nakagawa, Apr 24 2006
%E A117302 Edited by _G. C. Greubel_, May 07 2019