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A114208
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Number of permutations of [n] having exactly one fixed point and avoiding the patterns 123 and 231.
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3
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1, 0, 3, 2, 6, 6, 12, 10, 21, 16, 31, 24, 44, 32, 60, 42, 77, 54, 97, 66, 120, 80, 144, 96, 171, 112, 201, 130, 232, 150, 266, 170, 303, 192, 341, 216, 382, 240, 426, 266, 471, 294, 519, 322, 570, 352, 622, 384, 677, 416, 735, 450, 794, 486, 856, 522, 921, 560
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OFFSET
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1,3
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LINKS
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FORMULA
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n^2/6 if n mod 6 = 0; (7*n^2-12*n+29)/24 if n mod 6 = 1 or 5; (n^2-4)/6 if n mod 6 = 2 or 4; (7*n^2-12*n+45)/24 if n mod 6 = 3.
a(n) = a(n-1)+ 2*a(n-2)+3*a(n-3)-3*a(n-5)-2*a(n-6)+a(n-7)+a(n-8). [Harvey P. Dale, Mar 05 2012]
G.f.: -x*(2*x^6+2*x^5+2*x^4+2*x^3+x^2+x+1) / ((x-1)^3*(x+1)^3*(x^2+x+1)). [Colin Barker, Aug 11 2013]
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EXAMPLE
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a(2)=0 because none of the permutations 12 and 21 has exactly one fixed point.
a(3)=3 because we have 132, 213 and 321.
a(4)=2 because we have 4132 and 4213.
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MAPLE
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a:=proc(n) if n mod 6 = 0 then n^2/6 elif n mod 6 = 1 or n mod 6 = 5 then (7*n^2-12*n+29)/24 elif n mod 6 = 2 or n mod 6 = 4 then (n^2-4)/6 else (7*n^2-12*n+45)/24 fi end: seq(a(n), n=1..70);
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MATHEMATICA
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npn[n_]:=Module[{c=Mod[n, 6]}, Which[c==0, n^2/6, c==1, (7n^2-12n+29)/24, c==2, (n^2-4)/6, c==3, (7n^2-12n+45)/24, c==4, (n^2-4)/6, c==5, (7n^2-12n+29)/24]]; Array[npn, 60] (* or *) LinearRecurrence[{-1, 2, 3, 0, -3, -2, 1, 1}, {1, 0, 3, 2, 6, 6, 12, 10}, 60] (* Harvey P. Dale, Mar 05 2012 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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