OFFSET
1,1
COMMENTS
a(n) is the first Zagreb index of the Hanoi graph H[n].
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.
The M-polynomial of the Hanoi graph H[n] is M(H[n],x,y) = 6*x^2*y^3 + (3/2)*(3^n - 5)*x^3*y^3.
LINKS
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. (2004) Vol. 50, 83-92.
El-Mehdi Mehiri, Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs, arXiv:2511.12587 [math.CO], 2025. See p. 30, Table 12.
Eric. W. Weisstein's World of Mathematics, Hanoi Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).
FORMULA
O.g.f.: 6*x*(2 + 3*x)/((1 - x)*(1 - 3*x)).
E.g.f.: 3*(1 - exp(x))*(2 - 3*exp(x) - 3*exp(2*x)). - Bruno Berselli, Nov 14 2016
a(n) = 3*A168613(n+1). - R. J. Mathar, Apr 07 2022
MAPLE
seq(9*3^n-15, n = 1..30);
MATHEMATICA
Table[9 3^n - 15, {n, 1, 30}] (* Bruno Berselli, Nov 14 2016 *)
PROG
(Magma) [9*3^n-15: n in [1..30]]; // Bruno Berselli, Nov 14 2016
(PARI) a(n)=3^(n+2)-15 \\ Charles R Greathouse IV, Nov 14 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Nov 05 2016
STATUS
approved