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A243965
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Number of Dyck paths of semilength n such that both consecutive patterns of Dyck paths of semilength 2 occur at least once.
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3
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0, 0, 0, 0, 2, 10, 44, 179, 702, 2701, 10278, 38866, 146450, 550817, 2070116, 7779655, 29248932, 110047905, 414446256, 1562538171, 5898049688, 22290789562, 84351810044, 319609669957, 1212552963576, 4606078246284, 17518748817596, 66712192842068, 254346235738120
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OFFSET
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0,5
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COMMENTS
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The consecutive patterns 1010, 1100 are counted. Here 1=Up=(1,1), 0=Down=(1,-1).
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LINKS
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FORMULA
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EXAMPLE
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a(4) = 2: 10101100, 11001010.
a(5) = 10: 1010101100, 1010110010, 1010111000, 1011001010, 1100101010, 1100110100, 1101001100, 1101011000, 1110001010, 1110010100.
Here 1=Up=(1,1), 0=Down=(1,-1).
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MAPLE
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b:= proc(x, y, t, s) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, `if`(s={}, 1, 0), `if`(nops(s)>x, 0, add(
b(x-1, y-1+2*j, irem(2*t+j, 8), s minus {2*t+j}), j=0..1))))
end:
a:= n-> b(2*n, 0, 0, {10, 12}):
seq(a(n), n=0..30);
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MATHEMATICA
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b[x_, y_, t_, s_] := b[x, y, t, s] = If[y<0 || y>x, 0, If[x == 0, If[s == {}, 1, 0], If[Length[s] > x, 0, Sum[b[x - 1, y - 1 + 2 j, Mod[2t + j, 8], s ~Complement~ {2t + j}], {j, 0, 1}]]]];
a[n_] := b[2n, 0, 0, {10, 12}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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