cp's OEIS Frontend

A174085 Number of permutations of length n with no consecutive triples i,...i+r,...i+2r for all positive and negative r, and for all equal spacings d.

%I A174085 #26 May 18 2025 19:06:39
%S A174085 1,1,2,4,18,72,396,2328,17050,131764,1199368,11379524,123012492,
%T A174085 1386127700,17450444866,227152227940
%N A174085 Number of permutations of length n with no consecutive triples i,...i+r,...i+2r for all positive and negative r, and for all equal spacings d.
%C A174085 Here we count both the sequence 1,2,3 (r=1) as a progression in 1,2,3,0,4,5, (note d=1) and in 1,0,2,4,3,5 (here, d=2).
%C A174085 Number of permutations of 1..n with no 2-dimensional arithmetic progression of length 3: that is, no three points (i,p(i)), (j,p(j)) and (k,p(k)) such that j-i = k-j and p(j)-p(i) = p(k)-p(j). - _David Bevan_, Jun 16 2021
%F A174085 a(n) >= A003407(n) with equality only for n in {0, 1, 2, 3}.
%e A174085 a(3) = 4; 123 and 321 each contain a 3-term arithmetic progression.
%e A174085 Since the only possibilities for progressions for n=4 are d=1 and r=1 and -1, we get the same term as A095816(4).
%Y A174085 Cf. A095816, A174084, A174086, A174087.
%Y A174085 Cf. A179040 (number of permutations of 1..n with no three elements collinear).
%Y A174085 Cf. A003407 for another interpretation of avoiding 3-term APs.
%K A174085 nonn,more
%O A174085 0,3
%A A174085 _Isaac Lambert_, Apr 20 2010
%E A174085 a(0)-a(3) and a(10)-a(13) from _David Bevan_, Jun 16 2021
%E A174085 a(14)-a(15) from _Bert Dobbelaere_, May 18 2025