A295370 Number of permutations of [n] avoiding three consecutive terms in arithmetic progression.

1, 1, 2, 4, 18, 80, 482, 3280, 26244, 231148, 2320130, 25238348, 302834694, 3909539452, 54761642704, 816758411516, 13076340876500, 221396129723368, 3985720881222850, 75503196628737920, 1510373288335622576, 31634502738658957588, 696162960370556156224, 15978760340940405262668

Offset: 0

Author

Alois P. Heinz, Nov 20 2017

Keywords

• nonn

Comments

These are permutations of n whose second-differences are nonzero. - Gus Wiseman, Jun 03 2019

Examples

			a(3) = 4: 132, 213, 231, 312.
a(4) = 18: 1243, 1324, 1342, 1423, 2134, 2143, 2314, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4132, 4213, 4231, 4312.
		

Links

• Wikipedia, Arithmetic progression
• Index entries related to non-averaging sequences

Crossrefs

Column k=0 of A295390.

Cf. A003407, A174080, A238423, A238424, A338765.

Cf. A049988, A175342, A279945, A325545, A325849, A325850, A325851, A325874, A325875.

Programs

Maple

b:= proc(s, j, k) option remember; `if`(s={}, 1,
      add(`if`(k=0 or 2*j<>i+k, b(s minus {i}, i,
          `if`(2*i-j in s, j, 0)), 0), i=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..12);

Mathematica

Table[Length[Select[Permutations[Range[n]],!MemberQ[Differences[#,2],0]&]],{n,0,5}] (* Gus Wiseman, Jun 03 2019 *)
b[s_, j_, k_] := b[s, j, k] = If[s == {}, 1, Sum[If[k == 0 || 2*j != i + k, b[s~Complement~{i}, i, If[MemberQ[s, 2*i - j ], j, 0]], 0], {i, s}]];
a[n_] := a[n] = b[Range[n], 0, 0];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz *)

Extensions

a(22)-a(23) from Vaclav Kotesovec, Mar 22 2022