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

%I A304163 #28 Sep 12 2022 08:31:53
%S A304163 7,31,73,133,211,307,421,553,703,871,1057,1261,1483,1723,1981,2257,
%T A304163 2551,2863,3193,3541,3907,4291,4693,5113,5551,6007,6481,6973,7483,
%U A304163 8011,8557,9121,9703,10303,10921,11557,12211,12883,13573,14281
%N A304163 a(n) = 9*n^2 - 3*n + 1 with n>0.
%C A304163 a(n) provides the number of vertices in the HcDN1(n) network (see Fig. 3 in the Hayat et al. paper).
%H A304163 Colin Barker, <a href="/A304163/b304163.txt">Table of n, a(n) for n = 1..1000</a>
%H A304163 S. Hayat, M. A. Malik, and M. Imran, <a href="http://www.romjist.ro/content/cuprins18_2.html">Computing Topological Indices of Honeycomb Derived Networks</a>, Romanian Journal for Information Science and Technology, Volume 18, Number 2, 2015, pages 144-165.
%H A304163 Leo Tavares, <a href="/A304163/a304163.jpg">Illustration: Hexagonal Square Rays</a>
%H A304163 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A304163 From _Bruno Berselli_, May 10 2018: (Start)
%F A304163 O.g.f.:  x*(7 + 10*x + x^2)/(1 - x)^3.
%F A304163 E.g.f.: -1 + (1 + 3*x)^2*exp(x).
%F A304163 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A304163 a(n) = A003215(n-1) + 6*A000290(n). - _Leo Tavares_, Jul 21 2022
%e A304163 From _Andrew Howroyd_, May 09 2018: (Start)
%e A304163 Illustration of the order 1 graph:
%e A304163     o---o
%e A304163    / \ / \
%e A304163   o---o---o
%e A304163    \ / \ /
%e A304163     o---o
%e A304163 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.
%e A304163 (End)
%p A304163 seq(9*n^2-3*n+1, n = 1 .. 40);
%o A304163 (PARI) a(n) = 9*n^2-3*n+1; \\ _Altug Alkan_, May 09 2018
%o A304163 (PARI) Vec(x*(7 + 10*x + x^2)/(1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 23 2018
%o A304163 (Julia) [9*n^2-3*n+1 for n in 1:40] |> println # _Bruno Berselli_, May 10 2018
%Y A304163 Cf. A304164.
%Y A304163 First trisection of A002061 (without 1).
%Y A304163 Cf. A003215, A000290.
%K A304163 nonn,easy
%O A304163 1,1
%A A304163 _Emeric Deutsch_, May 09 2018