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A092186 a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1.

%I A092186 #26 Jun 10 2018 09:52:15
%S A092186 2,1,2,2,8,12,72,144,1152,2880,28800,86400,1036800,3628800,50803200,
%T A092186 203212800,3251404800,14631321600,263363788800,1316818944000,
%U A092186 26336378880000,144850083840000,3186701844480000,19120211066880000,458885065605120000,2982752926433280000
%N A092186 a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1.
%C A092186 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.
%C A092186 "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?
%C A092186 "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."
%C A092186 The solution to the first question is twice A002620(n-1); the solution to the second question is a(n).
%C A092186 Number of permutation of [n] with no pair of consecutive elements of the same parity. - _Vladeta Jovovic_, Nov 26 2007
%D A092186 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.
%D A092186 David Singmaster, Problem 1654, Mathematics Magazine 75 (October 2002). Solution in Mathematics Magazine 76 (October 2003).
%H A092186 Alois P. Heinz, <a href="/A092186/b092186.txt">Table of n, a(n) for n = 0..300</a>
%H A092186 David Singmaster, <a href="http://www.jstor.org/stable/3219173">Problem 1654</a>, Mathematics Magazine 75 p. 317 (October 2002). <a href="http://www.jstor.org/stable/3219092">Solution</a>, Mathematics Magazine 76 p. 321-322 (October 2003).
%H A092186 Shinji Tanimoto, <a href="http://arXiv.org/abs/math/0612135">Alternate Permutations and Signed Eulerian Numbers</a>, arXiv:math/0612135 [math.CO], 2006; Ann. Comb. 14 (2010), 355.
%p A092186 a:= proc(n) option remember; `if`(n<2, 2-n,
%p A092186       (n*(3*n-1)*(n-1)*a(n-2) -4*a(n-1))/(12*n-16))
%p A092186     end:
%p A092186 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 11 2013
%t A092186 f[n_] := If[EvenQ[n], 2 (n/2)!^2, ((n + 1)/2)! ((n - 1)/2)!]; Table[
%t A092186 f[n], {n, 0, 25}] (* _Geoffrey Critzer_, Aug 24 2013 *)
%Y A092186 Cf. A152877.
%Y A092186 Row sums of A125300. - _Alois P. Heinz_, Nov 18 2013
%K A092186 nonn
%O A092186 0,1
%A A092186 _N. J. A. Sloane_, based on correspondence from _Hugo Pfoertner_ and _Rob Pratt_, Apr 02 2004