OFFSET
0,3
COMMENTS
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. - R. J. Mathar, Jul 06 2009
LINKS
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. [Ross La Haye, Feb 22 2009]
Index entries for linear recurrences with constant coefficients, signature (6,-11,6)
FORMULA
a(n) = 3*StirlingS2(n,3) + StirlingS2(n,2) + 1.
a(n) = StirlingS2(n+1,3) + 1. - Ross La Haye, Jan 21 2008
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]
EXAMPLE
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}}.
MAPLE
f := n -> (1/2)*(3^n - 2^(n+1) + 3);
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ross La Haye, Jan 11 2008
EXTENSIONS
Edited by N. J. A. Sloane, Jul 06 2009
STATUS
approved