OFFSET
0,2
COMMENTS
For n>=3, a(n) is the first Zagreb index of the double-wheel graph DW[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 double-wheel graph DW[n] consists of two cycles C[n], whose vertices are connected to an additional vertex.
The M-polynomial of the double-wheel graph DW[n] is M(DW[n],x,y)= 2*n*x^3*y^3 + 2*n*x^3*y^{2*n}.
4*a(n) + 81 is a square. - Bruno Berselli, May 08 2018
LINKS
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
O.g.f.: 2*x*(11 - 7*x)/(1 - x)^3.
E.g.f.: 2*x*(11 + 2*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A139576(n).
EXAMPLE
a(3) = 90. Indeed, the double-wheel graph DW[3] has 6 edges with end-point degrees 3,3 and 6 edges with end-point degrees 3,6. Then the first Zagreb index is 6*6 + 6*9 = 90.
MAPLE
seq(4*n^2+18*n, n = 0..50);
MATHEMATICA
Table[4 n^2 + 18 n, {n, 0, 50}] (* Vincenzo Librandi, Nov 09 2016 *)
LinearRecurrence[{3, -3, 1}, {0, 22, 52}, 50] (* Harvey P. Dale, Mar 01 2022 *)
PROG
(Magma) [4*n^2+18*n: n in [0..50]]; // Vincenzo Librandi, Nov 09 2016
(PARI) a(n)=4*n^2+18*n \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Nov 08 2016
STATUS
approved