A083323 a(n) = 3^n - 2^n + 1.

1, 2, 6, 20, 66, 212, 666, 2060, 6306, 19172, 58026, 175100, 527346, 1586132, 4766586, 14316140, 42981186, 129009092, 387158346, 1161737180, 3485735826, 10458256052, 31376865306, 94134790220, 282412759266, 847255055012

Offset: 0

Author

Paul Barry, Apr 27 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A000225 (if this starts 1,1,3,7....).

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 0) 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 1) x = y. - Ross La Haye, Jan 10 2008

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009

Examples

			From _Gus Wiseman_, Dec 10 2019: (Start)
Also the number of achiral set-systems on n vertices, where a set-system is achiral if it is not changed by any permutation of the covered vertices. For example, the a(0) = 1 through a(3) = 20 achiral set-systems are:
  0  0    0           0
     {1}  {1}         {1}
          {2}         {2}
          {12}        {3}
          {1}{2}      {12}
          {1}{2}{12}  {13}
                      {23}
                      {123}
                      {1}{2}
                      {1}{3}
                      {2}{3}
                      {1}{2}{3}
                      {1}{2}{12}
                      {1}{3}{13}
                      {2}{3}{23}
                      {12}{13}{23}
                      {1}{2}{3}{123}
                      {12}{13}{23}{123}
                      {1}{2}{3}{12}{13}{23}
                      {1}{2}{3}{12}{13}{23}{123}
BII-numbers of these set-systems are A330217. Fully chiral set-systems are A330282, with covering case A330229.
(End)
		

Links

• M. H. Albert, M. D. Atkinson, and V. Vatter, Inflations of geometric grid classes: three case studies, arXiv:1209.0425 [math.CO], 2012.
• 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.
• Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A134319, A028243, A000079.

Cf. A000612, A003238, A330098, A330234.

Programs

GAP

List([0..30], n -> 3^n-2^n+1); # G. C. Greubel, Feb 13 2019

Magma

[3^n-2^n+1: n in [0..30]]; // G. C. Greubel, Feb 13 2019

Mathematica

LinearRecurrence[{6,-11,6}, {1,2,6}, 30] (* G. C. Greubel, Feb 13 2019 *)

PARI

a(n)=3^n-2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[3^n-2^n+1 for n in range(30)] # G. C. Greubel, Feb 13 2019

Formula

G.f.: (1-4*x+5*x^2)/((1-x)*(1-2*x)*(1-3*x)).

E.g.f.: exp(3*x) - exp(2*x) + exp(x).

Row sums of triangle A134319. - Gary W. Adamson, Oct 19 2007

a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 10 2008

a(n) = Sum_{k=0..n}(binomial(n,k)*A255047(k)). - Yuchun Ji, Feb 23 2019