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A116731
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Number of permutations of length n which avoid the patterns 321, 2143, 3124; or avoid the patterns 132, 2314, 4312, etc.
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7
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1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300, 2681, 3102, 3565, 4072, 4625, 5226, 5877, 6580, 7337, 8150, 9021, 9952, 10945, 12002, 13125, 14316, 15577, 16910, 18317, 19800, 21361, 23002, 24725, 26532
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OFFSET
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1,2
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COMMENTS
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Row sums of triangle A130154. Also, binomial transform of [1, 1, 2, 2, 0, 0, 0, ...]. - Gary W. Adamson, Oct 23 2007
Conjecture: also counts the distinct pairs (flips, iterations) that a bubble sort program generates when sorting all permutations of 1..n. - Wouter Meeussen, Dec 13 2008
a(n) is the number of lattice points (x,y) in the closed region bounded by the parabolas y = x*(x - n) and y = x*(n - x). - Clark Kimberling, Jun 01 2013
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LINKS
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Table of n, a(n) for n=1..44.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, arXiv:1302.2274 [math.CO], 2013.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Integers: Electronic Journal of Combinatorial Number Theory, 15 (2015), #A16.
Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)
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FORMULA
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G.f.: (3*x^2 - 2*x + 1)*x/(x - 1)^4.
a(n) = (n^3 - 3*n^2 + 5*n)/3. - Franklin T. Adams-Watters, Sep 13 2006
a(n) = A006527(n-1) + 1. - Vladimir Joseph Stephan Orlovsky, May 04 2011
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MATHEMATICA
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Table[(n^3-3*n^2+5*n)/3, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)
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CROSSREFS
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Cf. A006527, A130154.
Sequence in context: A096584 A002836 A116720 * A116722 A116730 A240847
Adjacent sequences: A116728 A116729 A116730 * A116732 A116733 A116734
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KEYWORD
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nonn,easy
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AUTHOR
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Lara Pudwell, Feb 26 2006
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EXTENSIONS
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More terms from Franklin T. Adams-Watters, Sep 13 2006
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STATUS
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approved
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