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%I #23 Feb 05 2018 09:36:50
%S 1,3,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,
%T 1027,1141,1261,1387,1519,1657,1801,1951,2107,2269,2437,2611,2791,
%U 2977,3169,3367,3571,3781,3997,4219,4447,4681,4921,5167,5419,5677,5941,6211,6487
%N The hex numbers (A003215) together with 3.
%C Apart from the term 3, these are precisely the sizes of hexagonal clusters of circles in the A_2 (or hexagonal) lattice.
%C These numbers include many common numbers of strands used in cables.
%H Colin Barker, <a href="/A298125/b298125.txt">Table of n, a(n) for n = 1..1000</a>
%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wire#Number_of_strands">Wire: Number of strands</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Colin Barker_, Feb 05 2018: (Start)
%F G.f.: x*(1 + x^2 + 6*x^3 - 2*x^4) / (1 - x)^3.
%F a(n) = 7 - 9*n + 3*n^2 for n>2.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
%F (End)
%o (PARI) Vec(x*(1 + x^2 + 6*x^3 - 2*x^4) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, Feb 05 2018
%Y Cf. A003215.
%K nonn,easy
%O 1,2
%A _Brian Watson_, Jan 13 2018
%E Entry revised by _N. J. A. Sloane_, Jan 22 2018
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