A304381 a(n) = 54*n^2 - 26*n + 4 (n>=1).

32, 168, 412, 764, 1224, 1792, 2468, 3252, 4144, 5144, 6252, 7468, 8792, 10224, 11764, 13412, 15168, 17032, 19004, 21084, 23272, 25568, 27972, 30484, 33104, 35832, 38668, 41612, 44664, 47824, 51092, 54468, 57952, 61544, 65244, 69052, 72968, 76992, 81124, 85364

Offset

1,1

Comments

a(n) is the second Zagreb index of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference.

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 O(n,n) is M(O(n,n); x,y) = 4*(n+1)x^2*y^2 + 8(n-1)x^2 *y^3 + (6n^2 - 10n+4)x^3*y^3.

More generally, the M-polynomial of O(m,n) is M(O(m,n); x,y) =2(m+n+2)x^2*y^2+4(m+n-2)x^2 *y^3+(6mn-5m-5n+4)x^3*y^3.

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.

M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10, 2016, 884-892.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

From Colin Barker, May 13 2018: (Start)

G.f.: 4*x*(8 + 18*x + x^2) / (1 - x)^3.

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

(End)

Maple

seq(54*n^2-26*n+4, n = 1 .. 40);

Mathematica

Table[54n^2-26n+4, {n, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {32, 168, 412}, 40] (* Harvey P. Dale, Mar 21 2020 *)

Prog

(PARI) Vec(4*x*(8 + 18*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 13 2018

Crossrefs

Cf. A304380.

Sequence in context: A197367 A297843 A172191 * A281467 A302621 A302413

Adjacent sequences: A304378 A304379 A304380 * A304382 A304383 A304384

Keyword

nonn,easy

Author

Emeric Deutsch, May 13 2018

Status

approved