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

0,4

Comments

Also permutations whose second differences have at least one zero.

Links

Table of n, a(n) for n=0..23.

Wikipedia, Arithmetic progression

Formula

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

Example

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)

Mathematica

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

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.

Sequence in context: A350971 A119692 A264152 * A337071 A354313 A351739

Adjacent sequences: A333628 A333629 A333630 * A333632 A333633 A333634

Keyword

nonn

Author

Gus Wiseman, Mar 31 2020

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

Status

approved