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A092186 a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1. 9
2, 1, 2, 2, 8, 12, 72, 144, 1152, 2880, 28800, 86400, 1036800, 3628800, 50803200, 203212800, 3251404800, 14631321600, 263363788800, 1316818944000, 26336378880000, 144850083840000, 3186701844480000, 19120211066880000, 458885065605120000, 2982752926433280000 (list; graph; refs; listen; history; text; internal format)



Singmaster's problem: "A salesman's office is located on a straight road. His n customers are all located along this road to the east of the office, with the office of customer k at distance k from the salesman's office. The salesman must make a driving trip whereby he leaves the office, visits each customer exactly once, then returns to the office.

"Because he makes a profit on his mileage allowance, the salesman wants to drive as far as possible during his trip. What is the maximum possible distance he can travel on such a trip and how many different such trips are there?

"Assume that if the travel plans call for the salesman to visit customer j immediately after he visits customer i, then he drives directly from i to j."

The solution to the first question is twice A002620(n-1); the solution to the second question is a(n).

Number of permutation of [n] with no pair of consecutive elements of the same parity. - Vladeta Jovovic, Nov 26 2007


A. O. Munagi, Alternating subsets and permutations, Rocky Mountain J. Math. 40 (6) (2010) 1965-1977 doi:10.1216/RJM-2010-40-6-1965, Corollary 3.2.

David Singmaster, Problem 1654, Mathematics Magazine 75 (October 2002). Solution in Mathematics Magazine 76 (October 2003).


Alois P. Heinz, Table of n, a(n) for n = 0..300

David Singmaster, Problem 1654, Mathematics Magazine 75 p. 317 (October 2002). Solution, Mathematics Magazine 76 p. 321-322 (October 2003).

Shinji Tanimoto, Alternate Permutations and Signed Eulerian Numbers, arXiv:math/0612135 [math.CO], 2006; Ann. Comb. 14 (2010), 355.


a:= proc(n) option remember; `if`(n<2, 2-n,

      (n*(3*n-1)*(n-1)*a(n-2) -4*a(n-1))/(12*n-16))


seq(a(n), n=0..30);  # Alois P. Heinz, Nov 11 2013


f[n_] := If[EvenQ[n], 2 (n/2)!^2, ((n + 1)/2)! ((n - 1)/2)!]; Table[

f[n], {n, 0, 25}] (* Geoffrey Critzer, Aug 24 2013 *)


Cf. A152877.

Row sums of A125300. - Alois P. Heinz, Nov 18 2013

Sequence in context: A240492 A110775 A229232 * A138262 A276990 A127510

Adjacent sequences:  A092183 A092184 A092185 * A092187 A092188 A092189




N. J. A. Sloane, based on correspondence from Hugo Pfoertner and Rob Pratt, Apr 02 2004



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Last modified December 9 23:02 EST 2019. Contains 329880 sequences. (Running on oeis4.)