A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

0, 0, 0, 2, 6, 40, 238, 1760, 14076, 131732, 1308670, 14678452, 176166906, 2317481348, 32416648496, 490915956484, 7846449011500, 134291298372632, 2416652824505150, 46141903780094080, 922528719841017424, 19456439433050482412, 427837767407051523776, 9873256397944571377332

Offset: 0

Author

Gus Wiseman, Mar 31 2020

Keywords

• nonn

Comments

Also permutations whose second differences have at least one zero.

Examples

			The a(3) = 2 and a(4) = 6 permutations:
  (1,2,3)  (1,2,3,4)
  (3,2,1)  (1,4,3,2)
           (2,3,4,1)
           (3,2,1,4)
           (4,1,2,3)
           (4,3,2,1)
		

Links

• Wikipedia, Arithmetic progression

Crossrefs

The complement is counted by A295370.

The version for prime indices is A333195.

Strict partitions with equal differences are A049980.

Partitions with equal differences are A049988.

Compositions without triples in arithmetic progression are A238423.

Partitions without triples in arithmetic progression are A238424.

Strict partitions without triples in arithmetic progression are A332668.

Cf. A000142, A007862, A175342, A325849, A325850.

Programs

Mathematica

Table[Select[Permutations[Range[n]],MatchQ[Differences[#],{_,x_,x_,_}]&]//Length,{n,0,8}]

Formula

a(n) = n! - A295370(n).

Extensions

a(11)-a(21) (using A295370) from Giovanni Resta, Apr 07 2020

a(22)-a(23) (using A295370) from Alois P. Heinz, Jan 27 2024