A304380 a(n) = 36*n^2 - 4*n (n>=1).

32, 136, 312, 560, 880, 1272, 1736, 2272, 2880, 3560, 4312, 5136, 6032, 7000, 8040, 9152, 10336, 11592, 12920, 14320, 15792, 17336, 18952, 20640, 22400, 24232, 26136, 28112, 30160, 32280, 34472, 36736, 39072, 41480, 43960, 46512, 49136, 51832, 54600, 57440

Offset

1,1

Comments

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

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 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.

Sequence found by reading the line from 32, in the direction 32, 136, ..., in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, May 13 2018

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.: 8*x*(4 + 5*x) / (1 - x)^3.

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

(End)

Maple

seq(36*n^2-4*n, n = 1 .. 40);

Prog

(PARI) Vec(8*x*(4 + 5*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 13 2018
(GAP) List([1..50], n->36*n^2-4*n); # Muniru A Asiru, May 13 2018

Crossrefs

Cf. A304381.

Sequence in context: A044364 A044745 A297641 * A297887 A298492 A298292

Adjacent sequences: A304377 A304378 A304379 * A304381 A304382 A304383

Keyword

nonn,easy

Author

Emeric Deutsch, May 13 2018

Status

approved