login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A033138 a(n) = floor(2^(n+2)/7). 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Here we let p = 3 to produce the above sequence, but p can be an arbitrary natural number. By letting p = 2, 4, 6, 7 we produce A000975, A083593, A195904 and A117302. We denote by U[p,n,m] the number of cases in which the first player in a game of Russian roulette gets killed 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)st 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 the (pt+1)st 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_{z=0..t} binomial(n-pz-1,m-1), where t = floor((n-m)/p). Let A[p,n] be the number of the cases in which the first player gets 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]. - Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui, Jun 04 2006

Partial sums of A077947. - Mircea Merca, Dec 28 2010

a(n+1) is the number of partitions of n into two kinds of part 1 and one kind of part 2. - Joerg Arndt, Mar 10 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 926

S. Klavzar, Structure of Fibonacci cubes: a survey, Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia; Preprint series Vol. 49 (2011), 1150 ISSN 2232-2094.

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

FORMULA

a(n) = 2*a(n-1) + a(n-3) - 2*a(n-4). -John W. Layman

G.f.: 1/((1-x^3)*(1-2*x)); a(n) = sum{k=0..floor(n/3), 2^(n-3*k)}; a(n) = Sum_{k=0..n} 2^k*( cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3 ). - Paul Barry, Apr 16 2005

a(n) = floor(2^(n+2)/7). - Gary Detlefs, Sep 06 2010

a(n) = floor((4*2^n - 1)/7) = ceiling((4*2^n - 4)/7) = round((4*2^n - 2)/7) = round((8*2^n - 5)/14); a(n) = a(n-3) + 2^(n-1), n>3. - Mircea Merca, Dec 28 2010

a(n) = 4/7*2^n - 5/21*cos(2/3*Pi*n) + 1/21*3^(1/2)*sin(2/3*Pi*n)-1/3. - Leonid Bedratyuk, May 13 2012

MAPLE

seq(iquo(2^n, 7), n=3..34); # Zerinvary Lajos, Apr 20 2008

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 = 3 to produce the above sequence, but this code can produce A000975, A083593, A195904, A117302 for p = 2, 4, 6, 7. *) Table[A[3, n, 1], {n, 1, 20}] (* Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui, Jun 04 2006 *)

PROG

(MAGMA) [Round((4*2^n-2)/7): n in [1..40]]; // Vincenzo Librandi, Jun 25 2011

(PARI) a(n)=2^(n+2)\7 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A000975, A083593, A195904, A117302, A023001, A111662, A077947.

Sequence in context: A253585 A119027 A145139 * A155803 A282909 A282825

Adjacent sequences:  A033135 A033136 A033137 * A033139 A033140 A033141

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Edited by Jeremy Gardiner, Oct 08 2011

New name (using formula form Gary Detlefs) form Joerg Arndt, Mar 10 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 20 03:34 EDT 2017. Contains 290823 sequences.