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A116722 Number of permutations of length n which avoid the patterns 312, 1324, 3421; or avoid the patterns 312, 1324, 2341, etc.

%I A116722 #22 Sep 08 2022 08:45:24
%S A116722 1,2,5,12,25,47,82,135,212,320,467,662,915,1237,1640,2137,2742,3470,
%T A116722 4337,5360,6557,7947,9550,11387,13480,15852,18527,21530,24887,28625,
%U A116722 32772,37357,42410,47962,54045,60692,67937,75815,84362,93615,103612,114392,125995
%N A116722 Number of permutations of length n which avoid the patterns 312, 1324, 3421; or avoid the patterns 312, 1324, 2341, etc.
%H A116722 Colin Barker, <a href="/A116722/b116722.txt">Table of n, a(n) for n = 1..1000</a>
%H A116722 Ângela Mestre, José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H A116722 Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/maple/webbook/bookmain.html">Systematic Studies in Pattern Avoidance</a>, 2005.
%H A116722 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A116722 G.f.: x*(1 - 3*x + 5*x^2 - 3*x^3 + x^5) / (1 - x)^5.
%F A116722 For n >= 2, a(n) = (n^4 - 6*n^3 + 47*n^2 - 114*n + 120)/24. - _Franklin T. Adams-Watters_, Sep 16 2006
%F A116722 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4. - _Colin Barker_, Oct 24 2017
%o A116722 (PARI) Vec(x*(1 - 3*x + 5*x^2 - 3*x^3 + x^5) / (1 - x)^5 + O(x^60)) \\ _Colin Barker_, Oct 24 2017
%o A116722 (Magma) [1] cat [(n^4 - 6*n^3 + 47*n^2 - 114*n + 120)/24 : n in [2..50]]; // _Wesley Ivan Hurt_, Mar 25 2020
%K A116722 nonn,easy
%O A116722 1,2
%A A116722 _Lara Pudwell_, Feb 26 2006
%E A116722 Extended beyond a(30) by _R. J. Mathar_, Aug 05 2008