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A134063
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a(n) = (1/2)*(3^n - 2^(n+1) + 3).
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0
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1, 1, 2, 7, 26, 91, 302, 967, 3026, 9331, 28502, 86527, 261626, 788971, 2375102, 7141687, 21457826, 64439011, 193448102, 580606447, 1742343626, 5228079451, 15686335502, 47063200807, 141197991026, 423610750291, 1270865805302, 3812664524767, 11438127792026
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OFFSET
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0,3
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COMMENTS
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Let P(A) be the power set of an n-element set A. Then a(n-1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 2) x = y.
The inverse binomial transform yields A033484 with another leading 1. [From R. J. Mathar, Jul 06 2009]
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LINKS
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FORMULA
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a(n) = 3*StirlingS2(n,3) + StirlingS2(n,2) + 1.
a(n) = 6 a(n-1)-11 a(n-2) +6 a(n-3) (n >= 3). Also a(n) = 4 a(n-1)-3 a(n-2)+ 2^{n-2} (n >= 3). - Tian-Xiao He (the(AT)iwu.edu), Jul 02 2009
G.f.: -(1-4*x+6*x^2)/((x-1)*(3*x-1)*(2*x-1)). a(n+1)-a(n)=A001047(n+1). [R. J. Mathar, Jul 06 2009]
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EXAMPLE
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a(3) = 7 because for P(A) = {{},{1},{2},{1,2}} we have: case 0 {{1},{2}}, case 1 {{1},{1,2}}, {2},{1,2}}, case 2 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}.
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MAPLE
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f := n -> (1/2)*(3^n - 2^(n+1) + 3);
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MATHEMATICA
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Table[(3^n-2^(n+1)+3)/2, {n, 0, 30}] (* or *) LinearRecurrence[{6, -11, 6}, {1, 1, 2}, 30] (* Harvey P. Dale, May 05 2020 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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