OFFSET
1,2
COMMENTS
Also the number of 2-block covers of a labeled n-set. a(n) = A055154(n,2). Generally, number of k-block covers of a labeled n-set is T(n,k) = (1/k!)*Sum_{i = 1..k + 1} Stirling1(k + 1,i)*(2^(i - 1) - 1)^n. In particular, T(n,2) = (1/2!)*(3^n - 3), T(n,3) = (1/3!)*(7^n - 6*3^n + 11), T(n,4) = (1/4)!*(15^n - 10*7^n + 35*3^n - 50), ... - Vladeta Jovovic, Jan 19 2001
Conjectured to be the number of integers from 0 to 10^(n-1) - 1 that lack 0, 1, 2, 3, 4, 5 and 6 as a digit. - Alexandre Wajnberg, Apr 25 2005. This is easily verified to be true. - Renzo Benedetti, Sep 25 2008
Number of monic irreducible polynomials of degree 1 in GF(3)[x1,...,xn]. - Max Alekseyev, Jan 23 2006
Also, the greatest number of identical weights among which an odd one can be identified and it can be decided if the odd one is heavier or lighter, using n weighings with a comparing balance. If the odd one only needs to be identified, the sequence starts 4, 13, 40 and is A003462 (3^n - 1)/2, n > 1. - Tanya Khovanova, Dec 11 2006; corrected by Samuel E. Rhoads, Apr 18 2016
Binomial transform yields A134057. Inverse binomial transform yields A062510 with one additional 0 in front. - R. J. Mathar, Jun 18 2008
Numbers k where the recurrence s(0)=0, if s(k-1) >= k then s(k) = s(k-1) - k otherwise s(k) = s(k-1) + k produces s(k) = 0. - Hugo Pfoertner, Jan 05 2012
For n > 1: A008344(a(n)) = a(n). - Reinhard Zumkeller, May 09 2012
Also the number of edges in the (n-1)-Hanoi graph. - Eric W. Weisstein, Jun 18 2017
A level 1 Sierpiński triangle graph is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles. a(n) is the number of degree 4 vertices in the level n Sierpinski triangle graph. - Allan Bickle, Jul 30 2020
Also the number of minimum vertex cuts in the n-Apollonian network. - Eric W. Weisstein, Dec 20 2020
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..300
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
A. Born, C. A. J. Hukrnes, and G. J. Woeginger, How to detect a counterfeit coin: adaptive versus non-adaptive solutions, Inf. Proc. Lett. 86 (2003) 137-141.
G. Darby, The Counterfeit Coin
L. Halbeisen and N. Hungerbuhler, The general counterfeit coin problem, Discr. Math 147 (1-3) (1995) 139-150, Theorem 1 with b=1.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
B. Manvel, Counterfeit coin problems, Math. Mag. 50 (2) (1977) 90-92, theorem 2.
A. Stenger and J. Wert, The Twelve Coins (or Twelve bags of Gold)
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Minimum Vertex Cut
Index entries for linear recurrences with constant coefficients, signature (4,-3).
FORMULA
a(n) = 3*a(n-1) + 3. - Alexandre Wajnberg, Apr 25 2005
O.g.f: 3*x^2/((1-x)*(1-3*x)). - R. J. Mathar, Jun 18 2008
a(n) = 3^(n-1) + a(n-1) (with a(1)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = 3*A003462(n-1). - R. J. Mathar, Sep 10 2015
E.g.f.: 3*(-1 + exp(2*x))*exp(x)/2. - Ilya Gutkovskiy, Apr 19 2016
a(n) = A067771(n-1) - 3. - Allan Bickle, Jul 30 2020
a(n) = sigma(A008776(n-2)) for n>=2. - Flávio V. Fernandes, Apr 20 2021
EXAMPLE
For the Sierpiński triangle, Level 1 is a triangle, so a(1) = 0.
Level 2 has three corners (degree 2) and three degree 4 vertices, so a(2) = 3.
The level 2 Hanoi graph has 3 triangles joined by 3 edges, so a(2+1) = 12.
MAPLE
a:=n->sum(3^j, j=1..n): seq(a(n), n=0..23); # Zerinvary Lajos, Jun 27 2007
MATHEMATICA
Table[(3^n - 3)/2, {n, 24}] (* Alonso del Arte, Dec 29 2014 *)
PROG
(Magma) [(3^n-3)/2: n in [1..30]]; // Vincenzo Librandi, Jun 05 2011
(PARI) a(n)=(3^n-3)\2 \\ Charles R Greathouse IV, Apr 17 2012
(Haskell)
a029858 = (`div` 2) . (subtract 3) . (3 ^)
a029858_list = iterate ((+ 3) . (* 3)) 0
-- Reinhard Zumkeller, May 09 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected by T. D. Noe, Nov 07 2006
STATUS
approved