OFFSET
1,1
COMMENTS
a(n) is the first Zagreb index of the HcDN1(n) network (see Fig. 3 in the Hayat et al. manuscript).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of HcDN1(n) is M(HcDN1(n);x,y) = 6x^3*y^3 + 12(n-1)x^3*y^5 + 6nx^3*y^6 + 18(n-1)x^5*y^6 + (27n^2 -57n +30)x^6*y^6. - Emeric Deutsch, May 11 2018
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
S. Hayat, M. A. Malik, and M. Imran, Computing topological indices of honeycomb derived networks, Romanian J. of Information Science and Technology, 18, No. 2, 2015, 144-165.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Colin Barker, May 10 2018: (Start)
G.f.: 6*x*(15 + 76*x + 17*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
E.g.f.: 6*(exp(x)*(17 - 2*x + 54*x^2) - 17). - Stefano Spezia, Apr 15 2023
MAPLE
seq(324*n^2-336*n+102, n=1..40);
MATHEMATICA
Table[324n^2-336n+102, {n, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {90, 726, 2010}, 40] (* Harvey P. Dale, Apr 12 2020 *)
PROG
(PARI) a(n) = 324*n^2-336*n+102; \\ Altug Alkan, May 09 2018
(PARI) Vec(6*x*(15 + 76*x + 17*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, May 10 2018
(GAP) List([1..40], n->324*n^2-336*n+102); # Muniru A Asiru, May 10 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 09 2018
STATUS
approved