%I #31 Sep 08 2022 08:45:24

%S 1,2,5,12,26,52,98,177,310,531,895,1491,2463,4044,6611,10774,17520,

%T 28446,46136,74771,121116,196117,317485,513877,831661,1345862,2177873,

%U 3524112,5702390,9226936,14929790,24157221,39087538,63245319,102333451,165579399,267913515

%N Number of permutations of length n which avoid the patterns 231, 1423, 3214.

%H Colin Barker, <a href="/A116717/b116717.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Dairyko, Lara Pudwell, Samantha Tyner, Casey Wynn, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p22">Non-contiguous pattern avoidance in binary trees</a>. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.

%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 (4,-5,1,2,-1).

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

%F a(n) = A000045(n+5) - A000124(n+2). - _Charlie Marion_ and _Lara Pudwell_, Jan 15 2014

%F From _Colin Barker_, Oct 20 2017: (Start)

%F a(n) = -3 + (2^(-1- n)*((1-sqrt(5))^n*(-11+5*sqrt(5)) + (1+sqrt(5))^n*(11+5*sqrt(5)))) / sqrt(5) - n - (1 + n)*(2 + n)/2.

%F a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5) for n>5.

%F (End)

%F a(n) = 1 + Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j-1} Fibonacci(i). - _Ehren Metcalfe_, Oct 22 2017

%t LinearRecurrence[{4, -5, 1, 2, -1}, {1, 2, 5, 12, 26}, 40] (* _Vincenzo Librandi_, Oct 22 2017 *)

%o (PARI) Vec(x*(1 - 2*x + 2*x^2 + x^3 - x^4) / ((1 - x)^3*(1 - x - x^2)) + O(x^40)) \\ _Colin Barker_, Oct 20 2017

%o (Magma) I:=[1,2,5,12,26]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Oct 22 2017

%K nonn,easy

%O 1,2

%A _Lara Pudwell_, Feb 26 2006