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%I #35 May 30 2016 04:39:30
%S 1,2,7,29,121,497,2017,8129,32641,130817,523777,2096129,8386561,
%T 33550337,134209537,536854529,2147450881,8589869057,34359607297,
%U 137438691329,549755289601,2199022206977,8796090925057,35184367894529
%N a(n) = 2^(n-1)*(2^n - 1) + 1.
%C 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.
%H G. C. Greubel, <a href="/A134169/b134169.txt">Table of n, a(n) for n = 0..1000</a>
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(n) = 2^(n-1)*(2^n - 1) + 1.
%F a(n) = StirlingS2(2^n,2^n - 1) + 1 = C(2^n,2) + 1 = A006516(n) + 1.
%F From _R. J. Mathar_, Feb 15 2010: (Start)
%F a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
%F G.f.: (1 - 5*x + 7*x^2)/((1-x) * (2*x-1) * (4*x-1)). (End)
%e 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.
%t Table[EulerE[2,2^n],{n,0,60}]/2+1 (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t LinearRecurrence[{7,-14,8},{1,2,7},30] (* _Harvey P. Dale_, Mar 12 2013 *)
%Y Cf. A000392, A032263, A028243, A000079, A006516.
%K nonn,easy
%O 0,2
%A _Ross La Haye_, Jan 12 2008
%E More terms from _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009
%E Edited by _N. J. A. Sloane_, Jan 25 2015 at the suggestion of _Michel Marcus_
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