OFFSET
0,2
COMMENTS
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 (Case 0) x and y are disjoint, x is not a subset of y, and y is not a subset of x; or (Case 1) x and y are intersecting, but x is not a subset of y, and y is not a subset of x; or (Case 2) x and y are intersecting, and either x is a proper subset of y, or y is a proper subset of x; or (Case 3) x = y.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).
FORMULA
a(n) = 2^(n-1)*(2^n - 1) + 1.
a(n) = StirlingS2(2^n,2^n - 1) + 1 = C(2^n,2) + 1 = A006516(n) + 1.
From R. J. Mathar, Feb 15 2010: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: (1 - 5*x + 7*x^2)/((1-x) * (2*x-1) * (4*x-1)). (End)
EXAMPLE
a(2) = 7 because for P(A) = {{},{1},{2},{1,2}} we have for Case 0 {{1},{2}}; we have for Case 2 {{1},{1,2}}, {{2},{1,2}}; and we have for Case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under Case 1.
MATHEMATICA
Table[EulerE[2, 2^n], {n, 0, 60}]/2+1 (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
LinearRecurrence[{7, -14, 8}, {1, 2, 7}, 30] (* Harvey P. Dale, Mar 12 2013 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ross La Haye, Jan 12 2008
EXTENSIONS
More terms from Vladimir Joseph Stephan Orlovsky, Nov 03 2009
Edited by N. J. A. Sloane, Jan 25 2015 at the suggestion of Michel Marcus
STATUS
approved