OFFSET
1,2
COMMENTS
a(n) is the first Zagreb index of the King graph K(n,n) (n>=3). 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 the King graph K(n,n) is M(K(n,n); x,y) = 8x^3*y^5 + 4x^3*y^8 + 4(n - 2)x^5*y^5 + 4(3n - 8)x^5*y^8 + 2(n-3)(2n-5)x^8*y^8.
More generally, the M-polynomial of the King graph K(m,n) is M(K(m,n); x,y) = 8x^3*y^5 + 4x^3*y^8 + 2(m + n - 4)x^5*y^5 + 2(3m + 3n - 16)x^5*y^8 + (4mn - 11m -11n + 30)x^8*y^8.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Eric Weisstein's World of Mathematics, King Graph
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Colin Barker, May 18 2018: (Start)
G.f.: 4*x^2*(9 + 23*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. (End)
MAPLE
seq(4*(n-1)*(16*n-23), n=1..40);
MATHEMATICA
Table[4(n-1)(16n-23), {n, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 36, 200}, 40] (* Harvey P. Dale, Jul 03 2020 *)
PROG
(PARI) concat(0, Vec(4*x^2*(9 + 23*x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, May 18 2018
(GAP) List([1..40], n-> 4*(n-1)*(16*n-23)); # Muniru A Asiru, May 22 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 12 2018
STATUS
approved