OFFSET
1,1
COMMENTS
a(n) is the second Zagreb index of the King graph K(n,n) (n>=3). The second Zagreb index of a simple connected graph is the sum of the degree products 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*(21 - 57*x + 164*x^2) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
(End)
MAPLE
seq(256*n^2 - 828*n + 656, n=1..40);
PROG
(PARI) Vec(4*x*(21 - 57*x + 164*x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 18 2018
(GAP) List([1..40], n->256*n^2-828*n+656); # Muniru A Asiru, May 22 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 12 2018
STATUS
approved