cp's OEIS Frontend

A114210 Number of derangements of [n] avoiding the patterns 123 and 231.

%I A114210 #14 Mar 27 2024 08:59:00
%S A114210 0,1,1,3,4,7,8,14,13,23,20,34,28,48,37,64,48,82,60,103,73,126,88,151,
%T A114210 104,179,121,209,140,241,160,276,181,313,204,352,228,394,253,438,280,
%U A114210 484,308,533,337,584,368,637,400,693,433,751,468,811,504,874,541,939
%N A114210 Number of derangements of [n] avoiding the patterns 123 and 231.
%H A114210 T. Mansour and A. Robertson, <a href="http://dx.doi.org/10.1007/s000260200013">Refined restricted permutations avoiding subsets of patterns of length three</a>, Annals of Combinatorics, 6, 2002, 407-418.
%H A114210 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (-1,2,3,0,-3,-2,1,1).
%F A114210 a(n) = binomial(n,2) + 1 - A114208(n) - A114209(n)
%F A114210 a(n) = (7n^2-18n+24)/24 if n mod 6 = 0; (n^2-1)/6 if n mod 6 = 1 or 5; (7n^2-18n+32)/24 if n mod 6 = 2 or 4; (n^2-3)/6 if n mod 6 = 3.
%F A114210 G.f.: -x^2*(x^6+x^5+2*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^3*(x+1)^3*(x^2+x+1)). - _Colin Barker_, Aug 14 2013
%e A114210 a(2)=1 because we have 21; a(3)=1 because we have 312; a(4)=3 because we have 2143, 4312 and 4321.
%p A114210 a:=proc(n) if n mod 6 = 0 then (7*n^2-18*n+24)/24 elif n mod 6 = 1 or n mod 6 = 5 then (n^2-1)/6 elif n mod 6 = 2 or n mod 6 = 4 then (7*n^2-18*n+32)/24 else (n^2-3)/6 fi end: seq(a(n),n=1..70);
%t A114210 LinearRecurrence[{-1,2,3,0,-3,-2,1,1},{0,1,1,3,4,7,8,14},60] (* _Harvey P. Dale_, Mar 04 2023 *)
%o A114210 (PARI) Vec(-x^2*(x^6+x^5+2*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^3*(x+1)^3*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Aug 14 2013
%Y A114210 Cf. A114208, A114209.
%K A114210 nonn,easy
%O A114210 1,4
%A A114210 _Emeric Deutsch_, Nov 17 2005