OFFSET
0,3
COMMENTS
For all n, a(n)=1*3^n-3*1^n+3*0^n-1*0^n [with 0^0=1] where powers are taken of triangular numbers and multiplied by binomial coefficients with alternating signs. - Henry Bottomley, Jan 05 2001
For n>=1, a(n) is the number of facets of the harmonic polytope. See Ardila and Escobar. - Michel Marcus, Jun 08 2020
For n >= 3, this is the number of acyclic orientations of the wheel graph of order n+1. - Peter Kagey, Oct 13 2020
Number of ternary strings of length n with at least 2 different digits. - Enrique Navarrete, Nov 20 2020
A level 1 Hanoi graph is a triangle. Level n+1 is formed from three copies of level n by adding edges between pairs of corner vertices of each pair of triangles. This graph represents the allowable moves in the Towers of Hanoi problem with n disks. a(n) is the number of degree 3 vertices in the level n Hanoi graph. - Allan Bickle, Aug 07 2024
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Federico Ardila and Laura Escobar, The harmonic polytope, arXiv:2006.03078 [math.CO], 2020.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Wheel Graph
Index entries for linear recurrences with constant coefficients, signature (4,-3).
FORMULA
For n>0, a(n) = 3^n-3 = 3*a(n-1)+6.
a(0)=0, a(1)=0, a(2)=6, a(n) = 4*a(n-1)-3*a(n-2). - Harvey P. Dale, Sep 29 2013
G.f.: 6*x^2 / ((1 - x)*(1 - 3*x)). - Colin Barker, Oct 14 2020
EXAMPLE
a(2)=6 since intervals a-a and b-b can be combined as a-ab-b, a-b-ab, ab-a-b, b-ab-a, b-a-ab, or ab-b-a.
The level 2 Hanoi graph has 9 vertices, 6 with degree 3, so a(2) = 6.
MATHEMATICA
Join[{0}, NestList[3#+6&, 0, 30]] (* or *) Join[{0}, LinearRecurrence[{4, -3}, {0, 6}, 30]] (* Harvey P. Dale, Sep 29 2013 *)
PROG
(PARI) concat([0, 0], Vec(6*x^2 / ((1 - x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Oct 14 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 03 2001
STATUS
approved