|
| |
|
|
A134063
|
|
a(n) = (1/2)*(3^n - 2^(n+1) + 3).
|
|
0
| |
|
|
1, 1, 2, 7, 26, 91, 302, 967, 3026, 9331, 28502, 86527, 261626, 788971, 2375102, 7141687, 21457826, 64439011, 193448102, 580606447, 1742343626, 5228079451, 15686335502, 47063200807, 141197991026, 423610750291, 1270865805302, 3812664524767, 11438127792026
(list; graph; refs; listen; history; internal format)
|
|
|
|
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. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 06 2009]
|
|
|
REFERENCES
| 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. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
|
|
|
FORMULA
| a(n) = 3*StirlingS2(n,3) + StirlingS2(n,2) + 1.
a(n) = StirlingS2(n+1,3) + 1. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 21 2008
a(n) = 6 a(n-1)-11 a(n-2) +6 a(n-3) (n >= 3) with a(0) = a(1) = 1, a(2) = 2. Also a(n) = 4 a(n-1)-3 a(n-2)+ 2^{n-2} (n >= 3) with a(0) = a(1) = 1. - 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). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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);
|
|
|
CROSSREFS
| Cf. A000392, A028243, A000079.
Sequence in context: A091145 A000697 A027417 * A087448 A188860 A129273
Adjacent sequences: A134060 A134061 A134062 * A134064 A134065 A134066
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2008
|
|
|
EXTENSIONS
| Edited by N. J. A. Sloane, Jul 06 2009
|
| |
|
|
