%I #59 May 18 2024 14:53:22

%S 0,2,10,38,130,422,1330,4118,12610,38342,116050,350198,1054690,

%T 3172262,9533170,28632278,85962370,258018182,774316690,2323474358,

%U 6971471650,20916512102,62753730610,188269580438,564825518530,1694510110022,5083597438930,15250926534518,45753048039010

%N First differences of A003063.

%C 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

%C 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

%C 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]

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

%H Vincenzo Librandi, <a href="/A056182/b056182.txt">Table of n, a(n) for n = 0..1000</a>

%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.

%H Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "A area".

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).

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

%F 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]

%F a(n) = A217764(n,3). - _Ross La Haye_, Mar 27 2013

%F a(n) = sum_{i=1..n} binomial(n, i) * 2^(n - i + 1). - _Wesley Ivan Hurt_, Feb 10 2014

%F a(n) = 2 * A001047(n). - _Wesley Ivan Hurt_, Feb 10 2014

%F E.g.f.: 2*exp(2*x)*(exp(x) - 1). - _Stefano Spezia_, May 18 2024

%p A056182:=n->2 * (3^n - 2^n); seq(A056182(n), n=0..30); # _Wesley Ivan Hurt_, Feb 10 2014

%t Table[ -((-1 + k)^(1-k+n)*(-1+k)!)+k^(-k+n)*k! /. k -> 3, {n, 3, 36} ]

%t Table[2 (3^n - 2^n), {n, 0, 30}] (* _Wesley Ivan Hurt_, Feb 10 2014 *)

%t CoefficientList[Series[2 x/((2 x - 1) (3 x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)

%t LinearRecurrence[{5,-6},{0,2},30] (* _Harvey P. Dale_, Sep 22 2015 *)

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

%Y A002783(n) - 1.

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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Aug 05 2000

%E More terms from _Wouter Meeussen_, Aug 05 2000