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A321019 Coordination sequence for 3-D tiling (Cairo tiling) X Z, with respect to a 5-valent point.

%I A321019 #10 Nov 11 2018 06:14:15
%S A321019 1,5,16,36,63,98,143,196,255,322,399,484,575,674,783,900,1023,1154,
%T A321019 1295,1444,1599,1762,1935,2116,2303,2498,2703,2916,3135,3362,3599,
%U A321019 3844,4095,4354,4623,4900,5183,5474,5775,6084,6399,6722,7055,7396,7743,8098,8463,8836,9215,9602,9999
%N A321019 Coordination sequence for 3-D tiling (Cairo tiling) X Z, with respect to a 5-valent point.
%C A321019 By (Cairo tiling) X Z is meant a stack of layers of the planar Cairo tiling at integer levels.
%H A321019 Colin Barker, <a href="/A321019/b321019.txt">Table of n, a(n) for n = 0..1000</a>
%H A321019 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F A321019 G.f.: (x^6-x^5-2*x^4-4*x^2-x-1)*(1+x)/((x^2+1)*(x-1)^3), which is the product of the g.f.'s for A296368 and A040000.
%F A321019 From _Colin Barker_, Nov 11 2018: (Start)
%F A321019 a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>5.
%F A321019 a(n) = (-2 + (-i)^(1+n) + i^(1+n) + 8*n^2) / 2 for n>2, where i=sqrt(-1).
%F A321019 (End)
%o A321019 (PARI) Vec((1 + x)*(1 + x + 4*x^2 + 2*x^4 + x^5 - x^6) / ((1 - x)^3*(1 + x^2)) + O(x^50)) \\ _Colin Barker_, Nov 11 2018
%Y A321019 Cf. A040000, A296368.
%K A321019 nonn,easy
%O A321019 0,2
%A A321019 _N. J. A. Sloane_, Nov 10 2018