A304378 a(n) = 4*(n - 1)*(16*n - 23) for n >= 1.

0, 36, 200, 492, 912, 1460, 2136, 2940, 3872, 4932, 6120, 7436, 8880, 10452, 12152, 13980, 15936, 18020, 20232, 22572, 25040, 27636, 30360, 33212, 36192, 39300, 42536, 45900, 49392, 53012, 56760, 60636, 64640, 68772, 73032, 77420, 81936, 86580, 91352, 96252

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

Cf. A304379.

Sequence in context: A255093 A017138 A339981 * A120465 A361209 A144523

Adjacent sequences: A304375 A304376 A304377 * A304379 A304380 A304381

Keyword

nonn,easy

Author

Emeric Deutsch, May 12 2018

Status

approved