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Number of permutations of length n which avoid the patterns 1342, 3214, 4213.
1

%I #22 Jan 16 2019 15:17:33

%S 1,2,6,21,73,239,738,2178,6220,17351,47595,128985,346492,924788,

%T 2456502,6502017,17164189,45219875,118954134,312559974,820560736,

%U 2152792187,5645155791,14797355181,38776269808,101590174424,266111693898,696979788213,1825297432705

%N Number of permutations of length n which avoid the patterns 1342, 3214, 4213.

%H Colin Barker, <a href="/A116768/b116768.txt">Table of n, a(n) for n = 1..1000</a>

%H D. Callan, T. Mansour, <a href="http://arxiv.org/abs/1705.00933">Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns</a>, arXiv:1705.00933 [math.CO] (2017), Table 2 No 85.

%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/maple/webbook/bookmain.html">Systematic Studies in Pattern Avoidance</a>, 2005.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-18,21,-11,2).

%F G.f.: x*(1 - 5*x + 10*x^2 - 6*x^3 + 3*x^4) / ((1 - x)^2*(1 - 2*x)*(1 - 3*x + x^2)).

%F a(n) = 2^(-1-n)*(-7*4^n+5*(3+sqrt(5))^n - sqrt(5)*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5)) + 3*2^(1+n)*n). - _Colin Barker_, Nov 02 2017

%F a(n) = 3*n + 5*Fibonacci(2*n - 1) - 7*2^(n - 1). - _Ehren Metcalfe_, Nov 08 2017

%t LinearRecurrence[{7,-18,21,-11,2},{1,2,6,21,73},40] (* _Harvey P. Dale_, Jan 16 2019 *)

%o (PARI) Vec(x*(1 - 5*x + 10*x^2 - 6*x^3 + 3*x^4) / ((1 - x)^2*(1 - 2*x)*(1 - 3*x + x^2)) + O(x^30)) \\ _Colin Barker_, Nov 02 2017

%K nonn,easy

%O 1,2

%A _Lara Pudwell_, Feb 26 2006