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A083323
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a(n) = 3^n - 2^n + 1.
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28
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1, 2, 6, 20, 66, 212, 666, 2060, 6306, 19172, 58026, 175100, 527346, 1586132, 4766586, 14316140, 42981186, 129009092, 387158346, 1161737180, 3485735826, 10458256052, 31376865306, 94134790220, 282412759266, 847255055012
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history;
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A000225 (if this starts 1,1,3,7....).
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) 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 1) x = y. - Ross La Haye, Jan 10 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
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LINKS
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FORMULA
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G.f.: (1-4*x+5*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: exp(3*x) - exp(2*x) + exp(x).
a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 10 2008
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EXAMPLE
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Also the number of achiral set-systems on n vertices, where a set-system is achiral if it is not changed by any permutation of the covered vertices. For example, the a(0) = 1 through a(3) = 20 achiral set-systems are:
0 0 0 0
{1} {1} {1}
{2} {2}
{12} {3}
{1}{2} {12}
{1}{2}{12} {13}
{23}
{123}
{1}{2}
{1}{3}
{2}{3}
{1}{2}{3}
{1}{2}{12}
{1}{3}{13}
{2}{3}{23}
{12}{13}{23}
{1}{2}{3}{123}
{12}{13}{23}{123}
{1}{2}{3}{12}{13}{23}
{1}{2}{3}{12}{13}{23}{123}
BII-numbers of these set-systems are A330217. Fully chiral set-systems are A330282, with covering case A330229.
(End)
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MATHEMATICA
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LinearRecurrence[{6, -11, 6}, {1, 2, 6}, 30] (* G. C. Greubel, Feb 13 2019 *)
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PROG
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(Sage) [3^n-2^n+1 for n in range(30)] # G. C. Greubel, Feb 13 2019
(GAP) List([0..30], n -> 3^n-2^n+1); # G. C. Greubel, Feb 13 2019
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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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