OFFSET
1,2
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 either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) 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 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
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 (6,-11,6).
FORMULA
a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)
MAPLE
MATHEMATICA
LinearRecurrence[{6, -11, 6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n, 1, 50}] (* G. C. Greubel, Oct 06 2017 *)
PROG
(PARI) a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015
(Magma) [(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic and Goran Kilibarda, Feb 28 2000
STATUS
approved