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%I #12 Nov 09 2013 22:31:23
%S 0,0,0,0,0,29,230,1537,11208,89657,724755,6010150,55305521,545054759,
%T 5504044595,59482056555,690974195737,8306302563795,104653460921783,
%U 1401318441726295,19525683104731681,282626170020405627,4296152288224050974,67974610534037861728
%N Number of permutations of [n] with exactly two (possibly overlapping) occurrences of some of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).
%C The counted patterns are: 1243, 1342, 2341 (=UUD), 1324, 1423, 2314, 2413, 3412 (=UDU), 2134, 3124, 4123 (=DUU).
%H Alois P. Heinz, <a href="/A228408/b228408.txt">Table of n, a(n) for n = 0..185</a>
%e a(5) = 29: 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 21354, 21453, 23145, 23415, 23514, 24135, 24513, 25134, 31254, 31452, 32451, 34125, 34512, 35124, 41253, 41352, 42351, 45123, 51243, 51342, 52341.
%e a(6) = 230: 123546, 123645, 124365, ..., 651243, 651342, 652341.
%e a(7) = 1537: 1234657, 1234756, 1235476, ..., 7651243, 7651342, 7652341.
%p b:= proc(u, o, t, c) option remember;
%p `if`(c<0, 0, `if`(u+o=0, `if`(c=0, 1, 0),
%p add(b(u+j-1, o-j, [2, 3, 3, 6, 6, 3][t],
%p `if`(t in [5, 6], c-1, c)), j=1..o)+
%p add(b(u-j, o+j-1, [4, 5, 5, 4, 4, 5][t],
%p `if`(t=3, c-1, c)), j=1..u)))
%p end:
%p a:= n-> add(b(j-1, n-j, 1, 2), j=1..n):
%p seq(a(n), n=0..25);
%Y Column k=2 of A231384.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Nov 09 2013
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