A305070 a(n) = 378*n^2 - 54*n (n>=1).

324, 1404, 3240, 5832, 9180, 13284, 18144, 23760, 30132, 37260, 45144, 53784, 63180, 73332, 84240, 95904, 108324, 121500, 135432, 150120, 165564, 181764, 198720, 216432, 234900, 254124, 274104, 294840, 316332, 338580, 361584, 385344, 409860, 435132, 461160, 487944, 515484, 543780, 572832, 602640, 633204

Offset

1,1

Comments

a(n) is the first Zagreb index of the silicate network SL(n), defined pictorially in the Javaid et al. reference (Fig. 1, where SL(2) is shown) or in Liu et al. reference (Fig. 3, where again SL(2) is shown).

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 SL(n) is M(SL(n);x,y) = 6*n*x^3*y^3 + (18*n^2 + 6*n)*x^3*y^6 + (18*n^2 - 12*n)*x^6*y^6 (n>=2).

14*a(n)/27 + 1; is a square. - Muniru A Asiru, May 27 2018

Links

Muniru A Asiru, Table of n, a(n) for n = 1..5000

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

M. Javaid and C. Y. Jung, M-polynomials and topological indices of silicate and oxide networks, International J. Pure and Applied Math., 115, No. 1, 2017, 129-152.

J.-B. Liu, S. Wang, C. Wang, and S. Hayat, Further results on computation of topological indices of certain networks, IET Control Theory Appl., 11, No. 13, 2017, 2065-2071.

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

Formula

From Colin Barker, May 26 2018: (Start)

G.f.: 108*x*(3 + 4*x) / (1 - x)^3.

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

(End)

Maple

seq(378*n^2 - 54*n, n = 1 .. 50);

Mathematica

Table[378n^2-54n, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 324, 1404}, 50] (* Harvey P. Dale, Jan 29 2023 *)

Prog

(PARI) Vec(108*x*(3 + 4*x) / (1 - x)^3 + O(x^50)) \\ Colin Barker, May 26 2018
(GAP) List([1..50], n->378*n^2-54*n); # Muniru A Asiru, May 27 2018

Crossrefs

Cf. A305071.

Sequence in context: A329613 A202094 A202486 * A229538 A250437 A064197

Adjacent sequences: A305067 A305068 A305069 * A305071 A305072 A305073

Keyword

nonn,easy

Author

Emeric Deutsch, May 25 2018

Status

approved