A056182 First differences of A003063.

0, 2, 10, 38, 130, 422, 1330, 4118, 12610, 38342, 116050, 350198, 1054690, 3172262, 9533170, 28632278, 85962370, 258018182, 774316690, 2323474358, 6971471650, 20916512102, 62753730610, 188269580438, 564825518530, 1694510110022, 5083597438930, 15250926534518, 45753048039010

Offset: 0

Author

N. J. A. Sloane, Aug 05 2000

Keywords

• nonn
• easy

Comments

Let V be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xVy if x is a proper subset of y or y is a proper subset of x. Then a(n) = |V|. - Ross La Haye, Dec 22 2006

With regard to the comment by Ross La Haye: For nonempty subsets see a(n+1) in A260217. - If "proper" is omitted see A027649. - For nonempty subsets with "proper" omitted see A091344. - Manfred Boergens, Sep 04 2023

It appears that a(n) is the number of permutations p of 1,..,(n+2) such that max[p(i+1)-p(i)] is 2. For example, for n=1, the permutations (1,3,2) and (2,1,3) and no others have the desired property, so a(1)=2. This approach gives values in agreement with all listed terms. [John W. Layman, Nov 09 2011]

In the terdragon curve, a(n-1) is the number of enclosed unit triangles in expansion level n. - Kevin Ryde, Oct 20 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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.
• Kevin Ryde, Iterations of the Terdragon Curve, see index "A area".
• Index entries for linear recurrences with constant coefficients, signature (5,-6).

Crossrefs

3rd column of A056151. Cf. A028243 (partial sums).

A002783(n) - 1.

a(n) = A293181(n+1,3).

Cf. A001047, A027649, A091344, A217764, A260217.

Programs

Maple

A056182:=n->2 * (3^n - 2^n); seq(A056182(n), n=0..30); # Wesley Ivan Hurt, Feb 10 2014

Mathematica

Table[ -((-1 + k)^(1-k+n)*(-1+k)!)+k^(-k+n)*k! /. k -> 3, {n, 3, 36} ]
Table[2 (3^n - 2^n), {n, 0, 30}] (* Wesley Ivan Hurt, Feb 10 2014 *)
CoefficientList[Series[2 x/((2 x - 1) (3 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{5,-6},{0,2},30] (* Harvey P. Dale, Sep 22 2015 *)

Formula

a(n) = 2 * (3^n - 2^n).

a(n) = 5*a(n-1)-6*a(n-2). G.f.: 2*x/((2*x-1)*(3*x-1)). [Colin Barker, Dec 10 2012]

a(n) = A217764(n,3). - Ross La Haye, Mar 27 2013

a(n) = sum_{i=1..n} binomial(n, i) * 2^(n - i + 1). - Wesley Ivan Hurt, Feb 10 2014

a(n) = 2 * A001047(n). - Wesley Ivan Hurt, Feb 10 2014

E.g.f.: 2*exp(2*x)*(exp(x) - 1). - Stefano Spezia, May 18 2024

Extensions

More terms from Wouter Meeussen, Aug 05 2000