

A195904


Base 2 digits are, in order, the first n terms of the periodic sequence with initial period 1,0,0,0,0,0.


5



1, 2, 4, 8, 16, 32, 65, 130, 260, 520, 1040, 2080, 4161, 8322, 16644, 33288, 66576, 133152, 266305, 532610, 1065220, 2130440, 4260880, 8521760, 17043521, 34087042, 68174084, 136348168, 272696336, 545392672, 1090785345, 2181570690, 4363141380, 8726282760
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OFFSET

1,2


COMMENTS

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 nchambers and mbullets. 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 (m1) bullets are in {2,3,...,n}. We have binomial[n1,m1]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[np1,m1]cases for this. We continue to calculate and the last is (t), where t = Floor[(nm)/ 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[npt 1,m1]cases for this. Therefore U[p,n,m] = Sum[binomial[npz1,m1], for z = 0 to t, where t = Floor[(nm)/p]. Let A[p,n] be the number of the cases that the first player gets killed when pplayer use a gun with nchambers 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, Jun 04 2006


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,1,2).


FORMULA

From Colin Barker, Jun 09 2013: (Start)
a(n) = floor(2^(n+5)/63).
G.f.: x /(2*x^7 x^6 2*x +1).
G.f.: x /((x1)*(x+1)*(2*x1)*(x^2x+1)*( x^2+x+1)). (End)


MATHEMATICA

U[p_, n_, m_, v_]:=Block[{t}, t=Floor[(1+pm+nv)/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, Jun 04 2006
Rest[CoefficientList[Series[x/(2*x^7  x^6  2*x + 1), {x, 0, 50}], x]] (* G. C. Greubel, Sep 28 2017 *)


PROG

(PARI) x='x+O('x^50); Vec(x/(2*x^7  x^6  2*x + 1)) \\ G. C. Greubel, Sep 28 2017


CROSSREFS

Cf. A000975, A033138, A083593, A117302.
Sequence in context: A180209 A275073 A264701 * A101333 A023421 A098051
Adjacent sequences: A195901 A195902 A195903 * A195905 A195906 A195907


KEYWORD

nonn,base,easy


AUTHOR

Jeremy Gardiner, Sep 25 2011


EXTENSIONS

More terms from Colin Barker, Jun 09 2013


STATUS

approved



