0,9

Very close to 2^n

Here we let p = 6 to produce the above sequence, but p can be an arbitrary natural number. By letting p = 2, 3, 4, 7 we produce A000975, A033138, A083593 and A117302. We denote by U[p,n,m] the number of the cases that the first player gets killed in a Russian roulette game when 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 are going to calculate the following (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 (p+1)th chamber and the rest of the bullets are in {p+2,..,n}. We have binomial[n-p-1,m-1]-cases for this. We continue to calculate, and the last is (t), where t = Floor[(n-m)/ p]. (t) The first gets killed when one bullet is in (pt+1)-th chamber and the remaining bullets are in {pt+2,...,n}. We have binomial[n-pt- 1,m-1]-cases for this. Therefore U[p,n,m] = Sum[binomial[n-pz-1,m-1], for z = 0 to t, where t = Floor[(n-m)/p]. Let A[p,n] be the number of the cases that the first player gets killed when p-player use a gun with n-chambers and the number of the bullets can be from 1 to n. Then A[p,n] = Sum[U[p,n,m], m = 1 to n]. - Ryohei Miyadera,Tomohide Hashiba,Yuta Nakagawa,Hiroshi Matsui (miyadera1272000(AT)yahoo.co.jp), Jun 04 2006

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 = 6 to produce the above sequence, but this code can produce A000975, A033138, A083593, A117302 for p = 2, 3, 4, 7.*)Table[A[6, n, 1], {n, 1, 20}] - Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui (miyadera1272000(AT)yahoo.co.jp), Jun 04 2006

Cf. A000975, A033138, A083593, A117302.

Sequence in context: A023422 A084638 A006211 this_sequence A023421 A098051 A084637

Adjacent sequences: A101330 A101331 A101332 this_sequence A101334 A101335 A101336

nonn

Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 24 2004

More terms from Ryohei Miyadera,Tomohide Hashiba,Yuta Nakagawa,Hiroshi Matsui (miyadera1272000(AT)yahoo.co.jp), Jun 04 2006