A192009 -id:A192009 - cp's OEIS frontend

A192008 Modified linear phone booth sequence: number of ways to occupy n phone booths in a row, one by one, each time picking a phone booth adjacent to the smallest number of previously occupied phone booths.

1, 2, 4, 8, 32, 96, 456, 2016, 11232, 61632, 419328, 2695680, 21358080, 161049600, 1433894400, 12429158400, 123511910400, 1202903654400, 13229501644800, 143113833676800, 1722282128179200, 20516624400384000, 268083853148160000, 3485314242772992000, 49167975665958912000

Offset: 1

Jens Voß, Jun 21 2011

nonn

			For n=4, the A192008(n) = 8 ways of picking the phones are (1, 3, 4, 2), (1, 4, 2, 3), (1, 4, 3, 2), (2, 4, 1, 3), (3, 1, 4, 2), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3).

Max Alekseyev, Table of n, a(n) for n = 1..100
Project Euler, Comfortable distance (Problem 364).
Jens Voß, Java class for generating A192008

Cf. A095236, A192009, A276657.

{ A192008(n) = my(r,k); r=0; for(v=0,2, forstep(m=lift(Mod(n-1-v,3)/2),(n-1-v)\2,3, k=(n-1-v-2*m)\3; r+=(m+k+1)!*binomial(m+k,m)*2^k*(k+v)!*(m+k)!*(1+(v==1)););); r; } \\ Max Alekseyev, Sep 11 2016

a(n) = Sum (m+k+1)!*binomial(m+k,m)*2^k*(k+v1+v2)!*(m+k)!, where the sum is taken over v1,v2 each from 0 to 1, and over nonnegative m,k such that 2*m+3*k = n-1-v1-v2. - Max Alekseyev, Sep 11 2016

More terms from João Batista Souza de Oliveira, Jul 09 2014

Terms a(20) onward from Max Alekseyev, Sep 11 2016

A276657 Number of ways to occupy n unlabeled phone booths in a circle one by one, each time picking a phone booth adjacent to the smallest number of previously occupied phone booths.

Original entry on oeis.org

1, 1, 2, 2, 8, 28, 72, 432, 1728, 9792, 56448, 372096, 2419200, 19526400, 144806400, 1310515200, 11338444800, 112903372800, 1102226227200, 12163505356800, 131369759539200, 1589020051046400, 18899570737152000, 247773823008768000, 3220159580209152000, 45535430530695168000

Offset: 1

Max Alekseyev, Sep 11 2016

nonn

Each phone booth has two adjacent ones, each of which may or may not be occupied. So, available phone booths may have from 0 to 2 adjacent ones that are occupied. Each time we pick a phone booth with the smallest number of those (0 is top priority, then 1, then 2).

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A192009 (labeled case).

Cf. A095236, A192008.

r[n_] := {ToRules[Reduce[m >= 0 && k >= 0 && 2 m + 3 k == n, {m, k}, Integers]]}; f[{m_, k_}] := (m + k - 1)!*Binomial[m + k, m]*2^k*k!*(m + k)!; a[n_] := Total[f /@ ({m, k} /. r[n])]; a[1] = 1; Array[a, 26] (* Jean-François Alcover, Sep 13 2016 *)

{ A276657(n) = my(r,k); if(n==1, return(1)); r=0; forstep(m=lift(Mod(n, 3)/2), n\2, 3, k=(n-2*m)\3; r+=(m+k-1)!*binomial(m+k, m)*2^k*k!*(m+k)!); r; }

a(n) = A192009(n) / n.

For n > 1, a(n) = Sum (m+k-1)!*binomial(m+k,m)*2^k*k!*(m+k)!, where the sum is taken over nonnegative m,k such that 2*m+3*k = n.

cp's OEIS Frontend

A192008 Modified linear phone booth sequence: number of ways to occupy n phone booths in a row, one by one, each time picking a phone booth adjacent to the smallest number of previously occupied phone booths.

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