A028243 a(n) = 3^(n-1) - 2^n + 1 (essentially Stirling numbers of second kind).

0, 0, 2, 12, 50, 180, 602, 1932, 6050, 18660, 57002, 173052, 523250, 1577940, 4750202, 14283372, 42915650, 128878020, 386896202, 1161212892, 3484687250, 10456158900, 31372671002, 94126401612, 282395982050, 847221500580, 2541731610602, 7625329049532

Offset: 1

Author

N. J. A. Sloane, Doug McKenzie (mckfam4(AT)aol.com)

Keywords

• nonn
• easy

Comments

For n >= 3, a(n) is equal to the number of functions f: {1,2,...,n-1} -> {1,2,3} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007

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 x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 02 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 x is not a subset of y and y is not a subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009

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 a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x and x and y are disjoint. Then a(n+2) = |R|. - Ross La Haye, Mar 19 2009

In the terdragon curve, a(n) is the number of triple-visited points in expansion level n. The first differences of this sequence (A056182) are the number of enclosed unit triangles since on segment expansion each unit triangle forms a new triple-visited point, and existing triple-visited points are unchanged. - Kevin Ryde, Oct 20 2020

a(n+1) is the number of ternary strings of length n that contain at least one 0 and one 1; for example, for n=3, a(4)=12 since the strings are the 3 permutations of 100, the 3 permutations of 110, and the 6 permutations of 210. - Enrique Navarrete, Aug 13 2021

From Sanjay Ramassamy, Dec 23 2021: (Start)

a(n+1) is the number of topological configurations of n points and n lines where the points lie at the vertices of a convex cyclic n-gon and the lines are the perpendicular bisectors of its sides.

a(n+1) is the number of 2n-tuples composed of n 0's and n 1's which have an interlacing signature. The signature of a 2n-tuple (v_1,...,v_{2n}) is the n-tuple (s_1,...,s_n) defined by s_i=v_i+v_{i+n}. The signature is called interlacing if after deleting the 1's, there are letters remaining and the remaining 0's and 2's are alternating. (End)

a(n+1) is the number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty proper subset of B. If either "nonempty" or "proper" is omitted then see A001047. If "nonempty" and "proper" are omitted then see A000244. - Manfred Boergens, Mar 28 2023

a(n) is the number of (n-1) X (n-1) nilpotent Boolean relation matrices with rank equal to 1. a(n) = A060867(n-1) - A005061(n-1) (since every rank 1 matrix is either idempotent or nilpotent). - Geoffrey Critzer, Jul 13 2023

For odd n > 3, a(n) is also the number of minimum vertex colorings in the (n-1)-prism graph. - Eric W. Weisstein, Mar 05 2024

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..2096
• Ovidiu Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.
• J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
• K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters, Volume: 50, Issue: 1, January 2 2014.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• 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.
• P. Melotti, S. Ramassamy and P. Thévenin, Points and lines configurations for perpendicular bisectors of convex cyclic polygons, arXiv:2003.11006 [math.CO], 2020.
• Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
• Kevin Ryde, Iterations of the Terdragon Curve, see index "T triple-visited points".
• Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
• Eric Weisstein's World of Mathematics, Prism Graph.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A000392, A008277, A163626, A056182 (first differences), A000244, A001047.

Programs

Magma

[3^(n-1) - 2*2^(n-1) + 1: n in [1..30]]; // G. C. Greubel, Nov 19 2017

Mathematica

Table[2 StirlingS2[n, 3], {n, 24}] (* or *)
Table[3^(n - 1) - 2*2^(n - 1) + 1, {n, 24}] (* or *)
Rest@ CoefficientList[Series[-2 x^3/(-1 + x)/(-1 + 3 x)/(-1 + 2 x), {x, 0, 24}], x] (* Michael De Vlieger, Sep 24 2016 *)

PARI

a(n) = 3^(n-1) - 2*2^(n-1) + 1 \\ G. C. Greubel, Nov 19 2017

Sage

[stirling_number2(i,3)*2 for i in range(1,30)] # Zerinvary Lajos, Jun 26 2008

Formula

a(n) = 2*S(n, 3) = 2*A000392(n). - Emeric Deutsch, May 02 2004

G.f.: -2*x^3/(-1+x)/(-1+3*x)/(-1+2*x) = -1/3 - (1/3)/(-1+3*x) + 1/(-1+2*x) - 1/(-1+x). - R. J. Mathar, Nov 22 2007

E.g.f.: (exp(3*x) - 3*exp(2*x) + 3*exp(x) - 1)/3. - Wolfdieter Lang, May 03 2017

E.g.f. with offset 0: exp(x)*(exp(x)-1)^2. - Enrique Navarrete, Aug 13 2021

a(n) = Sum_{k = 1..n-2} binomial(n-1, k) * (2^(n-k-1)-1). - Ocean Wong, Jan 03 2025