A296531 Number of non-averaging permutations of [n] with first element ceiling(n/2).

1, 1, 1, 2, 3, 6, 13, 32, 51, 76, 161, 386, 903, 2280, 5018, 12828, 19720, 27656, 48788, 100120, 220686, 537208, 1258242, 3123166, 7056165, 17189752, 35968308, 82137764, 189847917, 509880208, 1322092262, 3807727932, 5678509066, 7721623440, 13293899416, 23650787296

Offset: 0

Author

Alois P. Heinz, Dec 14 2017

Keywords

• nonn

Comments

A non-averaging permutation avoids any 3-term arithmetic progression.

a(0) = 1 by convention.

Examples

			a(5) = 6: 31254, 31524, 31542, 35124, 35142, 35412.
a(6) = 13: 312564, 315264, 315426, 315462, 315624, 351264, 351426, 351462, 351624, 354126, 354162, 354612, 356124.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..99
• Eric Weisstein's World of Mathematics, Nonaveraging Sequence
• Wikipedia, Arithmetic progression
• Index entries related to non-averaging sequences

Crossrefs

Cf. A003407, A292523, A296529.

Programs

Maple

b:= proc(s) option remember; local n, r, ok, i, j, k;
      if nops(s) = 1 then 1
    else n, r:= max(s), 0;
         for j in s minus {n} do ok, i, k:= true, j-1, j+1;
           while ok and i>=0 and k b({$0..n} minus {ceil(n/2)-1}):
seq(a(n), n=0..25);

Mathematica

b[s_] := b[s] = Module[{n = Max[s], r = 0, ok, i, j, k}, If[Length[s] == 1, 1, Do[{ok, i, k} = {True, j - 1, j + 1}; While[ok && i >= 0 && k < n, {ok, i, k} = {FreeQ[s, i] ~Xor~ MemberQ[s, k], i - 1, k + 1}]; r = r + If[ok, b[s ~Complement~ {j}], 0], {j, s ~Complement~ {n}}]; r]];
a[n_] := b[Complement[Range[0, n], {Ceiling[n/2] - 1}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A296529(n,ceiling(n/2)).