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%I #28 Feb 19 2024 14:18:02
%S 1,1,2,4,18,92,570,4082,33292,304490,3086890,34357812,416526730,
%T 5463479106,77094352076,1164544912938,18749754351338,320544941916628,
%U 5799226664694602,110695180631374114,2223242026407894732,46868311165318977130,1034758905785710599402
%N Number of permutations of 1..n with no three elements in correct or reverse order.
%C Counts permutations with the property that no subsequence i(i+1)(i+2) or (i+2)(i+1)i occurs.
%H Andrew Howroyd, <a href="/A095816/b095816.txt">Table of n, a(n) for n = 0..200</a>
%H W. M. Dymacek and I. Lambert, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Dymacek/dymacek5.html">Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2</a>, J. Int. Seq. 14 (2011) # 11.1.6, Table 1.
%H D. M. Jackson and R. C. Read, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002031183">A note on permutations without runs of given length</a>, Aequationes Math. 17 (1978), no. 2-3, 336-343.
%F G.f. Sum_{n>=0} n!*((2*x^m-x^(m+1)-x)/(x^m-1))^n where m = 3. - Ivana Jovovic ( ivana121(AT)EUnet.yu ), Nov 11 2007
%F From _Vaclav Kotesovec_, May 26 2023: (Start)
%F a(n) ~ n! * (1 - 2/n + 6/n^2 - 28/(3*n^3) - 10/(3*n^4) + 496/(15*n^5) + 1384/(45*n^6) - 79724/(315*n^7) - 259306/(315*n^8) + 3718094/(2835*n^9) + 33233992/(2025*n^10) + ...).
%F a(n) = (n-3)*a(n-1) + 3*(n-1)*a(n-2) + (2*n-5)*a(n-3) - (n-3)*a(n-4) - (2*n-13)*a(n-5) - (n-8)*a(n-6) + (n-6)*a(n-7).
%F (End)
%o (PARI) seq(n)={my(m=3); Vec(sum(k=0, n, k!*((2*x^m-x^(m+1)-x)/(x^m-1) + O(x*x^n))^k))} \\ _Andrew Howroyd_, Aug 31 2018
%Y Cf. A002464, A095817, A095818.
%Y Cf. A165963, A165964, A078628. [From _Isaac Lambert_, Oct 07 2009]
%K nonn
%O 0,3
%A _Jonas Wallgren_, Jun 08 2004
%E More terms from Ivana Jovovic (ivana121(AT)EUnet.yu), Nov 11 2007
%E a(0)=1 prepended by _Max Alekseyev_, Jun 14 2011
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