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%I #16 Jan 11 2020 15:57:47
%S 6,14,32,50,82,114,164,214,286,358,456,554,682,810,972,1134,1334,1534,
%T 1776,2018,2306,2594,2932,3270,3662,4054,4504,4954,5466,5978,6556,
%U 7134,7782,8430,9152,9874,10674,11474,12356,13238,14206,15174,16232,17290,18442
%N The crystallogen sequence (a(n) = A018227(n)-4).
%C Terms up to 114 are the atomic numbers of the elements of group 14 in the periodic table. Those elements are also called crystallogens.
%H Colin Barker, <a href="/A271996/b271996.txt">Table of n, a(n) for n = 2..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Carbon_group">Carbon group</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F From _Colin Barker_, Jun 19 2016: (Start)
%F a(n) = (6*(-9 + (-1)^n) + (25 + 3*(-1)^n)*n + 12*n^2 + 2*n^3)/12.
%F a(n) = (n^3 + 6*n^2 + 14*n - 24)/6 for n even.
%F a(n) = (n^3 + 6*n^2 + 11*n - 30)/6 for n odd.
%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>7.
%F G.f.: 2*x^2*(3 + x - x^2 - 2*x^3 + x^5) / ((1-x)^4*(1+x)^2).
%F (End)
%t Table[(6*(-9+(-1)^n)+(25+3*(-1)^n)*n+12*n^2+2*n^3)/12, {n, 2, 10}] (* or *) LinearRecurrence[{2, 1, -4, 1, 2, -1}, {6, 14, 32, 50, 82, 114}, 50] (* _G. C. Greubel_, Jun 23 2016 *)
%o (PARI) Vec(2*x^2*(3+x-x^2-2*x^3+x^5)/((1-x)^4*(1+x)^2) + O(x^100)) \\ _Colin Barker_, Jun 19 2016
%Y Cf. A173592, A018227.
%K nonn,easy
%O 2,1
%A _Natan Arie Consigli_, Jun 18 2016
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