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A304201 a(n) is the number of cyclic permutations of length n that admit a [1,-1,-1]-gridding.

%I A304201 #23 Jul 01 2018 05:45:26
%S A304201 1,1,1,2,5,15,43,120,338,952,2672,7494,21035,59115,166433,469560,
%T A304201 1327802,3763545,10692500,30447858,86894361,248506757,712109663,
%U A304201 2044402512,5879579540,16937048040,48864612668,141179970820,408444645375,1183143522435,3431241484223,9961919944284
%N A304201 a(n) is the number of cyclic permutations of length n that admit a [1,-1,-1]-gridding.
%C A304201 a(n) is the number of permutations of length n that are composed of an increasing segment, followed by a decreasing segment, followed by another decreasing segment. In other words, these permutation have a descent set of the form {i, i+1, ..., n-1} for some i or {i, i+1, ..., n-1}\{j} for some i and j > i.
%F A304201 a(n) = A027376(n)/2 - Sum_{i=2..n-1} (n+1-i)*A303979(n,i), when n is odd and n > 2;
%F A304201 a(n) = (A027376(n) + A133267(n/2))/2 - Sum_{i=2..n-1} (n+1-i)*A303979(n,i), when n = 2 (mod 4) and n > 2.
%F A304201 a(n) = (A027376(n) + A006575(n/2))/2 - Sum_{i=2..n-1} (n+1-i)*A303979(n,i), when n = 0 (mod 4) and n > 2.
%o A304201 (PARI) t051168(n, k) = if (n==0, 1, (1/n) * sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d)));
%o A304201 T303979(n, k) = my(t=sum(j=1, k-1, (-1)^(k+j+1)*t051168(n, j))); if (!(n % 2), t += (-1)^(k+1)*sum(j=1, k-1, if (((n-j) % 4) == 2, t051168(n/2, j/2)))); t;
%o A304201 a027376(n) = if(n<1, n==0, sumdiv(n, d, moebius(n/d)*3^d)/n);
%o A304201 a133267(n) = sumdiv(n, d, moebius(d)*3^(n/d)/n - if (d%2, moebius(d)*(3^(n/d)-1)/(2*n)));
%o A304201 a006575(n) = sumdiv(n, d, if ( bitand(d, 1), moebius(d) * (3^(n/d)-1) , 0 ) ) / (2*n);
%o A304201 a(n) = if (n <= 2, 1, res = a027376(n)/2 - sum(i=2, n-1, (n+1-i)*T303979(n,i)); if (!(n%2), if ((n%4)==2, res += a133267(n/2)/2, res += a006575(n/2)/2)); res); \\ _Michel Marcus_, May 18 2018
%Y A304201 Cf. A027376, A303979, A133267.
%K A304201 nonn
%O A304201 0,4
%A A304201 _Kassie Archer_, May 08 2018
%E A304201 More terms from _Michel Marcus_, May 19 2018