A304378 - OEIS

%I #24 Feb 16 2025 08:33:54

%S 0,36,200,492,912,1460,2136,2940,3872,4932,6120,7436,8880,10452,12152,

%T 13980,15936,18020,20232,22572,25040,27636,30360,33212,36192,39300,

%U 42536,45900,49392,53012,56760,60636,64640,68772,73032,77420,81936,86580,91352,96252

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

%C 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.

%C 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.

%C 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.

%H Colin Barker, <a href="/A304378/b304378.txt">Table of n, a(n) for n = 1..1000</a>

%H E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Colin Barker_, May 18 2018: (Start)

%F G.f.: 4*x^2*(9 + 23*x)/(1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. (End)

%p seq(4*(n-1)*(16*n-23), n=1..40);

%t Table[4(n-1)(16n-23),{n,40}] (* or *) LinearRecurrence[{3,-3,1},{0,36,200},40] (* _Harvey P. Dale_, Jul 03 2020 *)

%o (PARI) concat(0, Vec(4*x^2*(9 + 23*x) / (1 - x)^3 + O(x^40))) \\ _Colin Barker_, May 18 2018

%o (GAP) List([1..40], n-> 4*(n-1)*(16*n-23)); # _Muniru A Asiru_, May 22 2018

%Y Cf. A304379.

%K nonn,easy,changed

%O 1,2

%A _Emeric Deutsch_, May 12 2018