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 A114210 Number of derangements of [n] avoiding the patterns 123 and 231. 2

%I

%S 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 104,179,121,209,140,241,160,276,181,313,204,352,228,394,253,438,280,

%U 484,308,533,337,584,368,637,400,693,433,751,468,811,504,874,541,939

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

%C a(n)=binomial(n,2)+1-A114208(n)-A114209(n)

%D T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (-1,2,3,0,-3,-2,1,1).

%F (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 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 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 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);

%o (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 Cf. A114208, A114209.

%K nonn,easy

%O 1,4

%A _Emeric Deutsch_, Nov 17 2005

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Last modified January 28 00:29 EST 2022. Contains 350654 sequences. (Running on oeis4.)