A304163 a(n) = 9*n^2 - 3*n + 1 with n>0.

7, 31, 73, 133, 211, 307, 421, 553, 703, 871, 1057, 1261, 1483, 1723, 1981, 2257, 2551, 2863, 3193, 3541, 3907, 4291, 4693, 5113, 5551, 6007, 6481, 6973, 7483, 8011, 8557, 9121, 9703, 10303, 10921, 11557, 12211, 12883, 13573, 14281

Offset: 1

Author

Emeric Deutsch, May 09 2018

Keywords

• nonn
• easy

Comments

a(n) provides the number of vertices in the HcDN1(n) network (see Fig. 3 in the Hayat et al. paper).

Examples

			From _Andrew Howroyd_, May 09 2018: (Start)
Illustration of the order 1 graph:
    o---o
   / \ / \
  o---o---o
   \ / \ /
    o---o
The order 2 graph is composed of 7 such hexagons and in general the HcDN1(n) graph is constructed from a honeycomb graph with each hexagon subdivided into triangles.
(End)
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• S. Hayat, M. A. Malik, and M. Imran, Computing Topological Indices of Honeycomb Derived Networks, Romanian Journal for Information Science and Technology, Volume 18, Number 2, 2015, pages 144-165.
• Leo Tavares, Illustration: Hexagonal Square Rays
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A304164.

First trisection of A002061 (without 1).

Cf. A003215, A000290.

Programs

Julia

[9*n^2-3*n+1 for n in 1:40] |> println # Bruno Berselli, May 10 2018

Maple

seq(9*n^2-3*n+1, n = 1 .. 40);

PARI

a(n) = 9*n^2-3*n+1; \\ Altug Alkan, May 09 2018

PARI

Vec(x*(7 + 10*x + x^2)/(1 - x)^3 + O(x^40)) \\ Colin Barker, May 23 2018

Formula

From Bruno Berselli, May 10 2018: (Start)

O.g.f.: x*(7 + 10*x + x^2)/(1 - x)^3.

E.g.f.: -1 + (1 + 3*x)^2*exp(x).

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

a(n) = A003215(n-1) + 6*A000290(n). - Leo Tavares, Jul 21 2022