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 A134169 a(n) = 2^(n-1)*(2^n - 1) + 1. 7
 1, 2, 7, 29, 121, 497, 2017, 8129, 32641, 130817, 523777, 2096129, 8386561, 33550337, 134209537, 536854529, 2147450881, 8589869057, 34359607297, 137438691329, 549755289601, 2199022206977, 8796090925057, 35184367894529 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000392, A032263, A028243, A000079, A006516. Sequence in context: A278815 A263367 A120757 * A052961 A150662 A278391 Adjacent sequences:  A134166 A134167 A134168 * A134170 A134171 A134172 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

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