A174080 Number of permutations of length n with no consecutive triples i,i+d,i+2d for all d>0.

1, 1, 2, 5, 21, 100, 597, 4113, 32842, 292379, 2925367, 31983248, 383514347, 4966286235, 69508102006, 1039315462467, 16627618496319, 282023014602100, 5075216962675445, 96263599713301975, 1925002914124917950

Offset: 0

Author

Isaac Lambert, Mar 15 2010

Keywords

• nonn
• more

Examples

			For n=4, there are 4!-a(4)=3 permutations with some consecutive triple i,i+d,i+2d. Note for n=4, only d=1 applies. Hence those three permutations are (0,1,2,3), (1,2,3,0), and (3,0,1,2). Since here only d=1, this is the same value of a(4) in A002628.
		

Crossrefs

Cf. A002628, A174072, A174081, A174082, A174083, A295370.

Programs

Maple

b:= proc(s, x, y) option remember; `if`(s={}, 1, add(
      `if`(x=0 or xy-j,
         b(s minus {j}, y, j), 0), j=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Apr 13 2021

Mathematica

b[s_, x_, y_] := b[s, x, y] = If[s == {}, 1, Sum[If[x == 0 || x < y || x-y != y-j, b[s~Complement~{j}, y, j], 0], {j, s}]];
a[n_] := b[Range[n], 0, 0];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Sep 27 2022, after Alois P. Heinz *)

Extensions

a(0)-a(3) and a(10)-a(20) from Alois P. Heinz, Apr 13 2021