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

%I A333631 #13 Jan 27 2024 14:24:51
%S A333631 0,0,0,2,6,40,238,1760,14076,131732,1308670,14678452,176166906,
%T A333631 2317481348,32416648496,490915956484,7846449011500,134291298372632,
%U A333631 2416652824505150,46141903780094080,922528719841017424,19456439433050482412,427837767407051523776,9873256397944571377332
%N A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.
%C A333631 Also permutations whose second differences have at least one zero.
%H A333631 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>
%F A333631 a(n) = n! - A295370(n).
%e A333631 The a(3) = 2 and a(4) = 6 permutations:
%e A333631   (1,2,3)  (1,2,3,4)
%e A333631   (3,2,1)  (1,4,3,2)
%e A333631            (2,3,4,1)
%e A333631            (3,2,1,4)
%e A333631            (4,1,2,3)
%e A333631            (4,3,2,1)
%t A333631 Table[Select[Permutations[Range[n]],MatchQ[Differences[#],{___,x_,x_,___}]&]//Length,{n,0,8}]
%Y A333631 The complement is counted by A295370.
%Y A333631 The version for prime indices is A333195.
%Y A333631 Strict partitions with equal differences are A049980.
%Y A333631 Partitions with equal differences are A049988.
%Y A333631 Compositions without triples in arithmetic progression are A238423.
%Y A333631 Partitions without triples in arithmetic progression are A238424.
%Y A333631 Strict partitions without triples in arithmetic progression are A332668.
%Y A333631 Cf. A000142, A007862, A175342, A325849, A325850.
%K A333631 nonn
%O A333631 0,4
%A A333631 _Gus Wiseman_, Mar 31 2020
%E A333631 a(11)-a(21) (using A295370) from _Giovanni Resta_, Apr 07 2020
%E A333631 a(22)-a(23) (using A295370) from _Alois P. Heinz_, Jan 27 2024