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A309118 Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.

%I A309118 #19 Sep 08 2022 08:46:21
%S A309118 6,6,12,18,24,24,36,30,48,36,60,42,72,48,84,54,96,60,108,66,120,72,
%T A309118 132,78,144,84,156,90,168,96,180,102,192,108,204,114,216,120,228,126,
%U A309118 240,132,252,138,264,144,276,150,288,156,300,162,312,168,324,174,336
%N A309118 Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.
%H A309118 Colin Barker, <a href="/A309118/b309118.txt">Table of n, a(n) for n = 1..1000</a>
%H A309118 Felix Fröhlich, <a href="/A309118/a309118.pdf">Layers in the rhombille tiling</a>, 2019.
%H A309118 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rhombille_tiling">Rhombille tiling</a>
%H A309118 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A309118 a(2*n+1) = A008594(n).
%F A309118 a(2*n) = A008588(n+1) for n > 1.
%F A309118 From _Colin Barker_, Jul 13 2019: (Start)
%F A309118 G.f.: 6*x*(1 + x + x^3 + x^4 - x^5) / ((1 - x)^2*(1 + x)^2).
%F A309118 a(n) = 2*a(n-2) - a(n-4) for n>6.
%F A309118 (End)
%e A309118 See illustration in Fröhlich, 2019.
%t A309118 Join[{6, 6}, LinearRecurrence[{0, 2, 0, -1}, {12, 18, 24, 24}, 60]] (* _Vincenzo Librandi_, Jul 16 2019 *)
%o A309118 (PARI) a(n) = if(n<3, 6, if(n%2==0, 6*((n+2)/2), 12*((n-1)/2)))
%o A309118 (PARI) Vec(6*x*(1 + x + x^3 + x^4 - x^5) / ((1 - x)^2*(1 + x)^2) + O(x^40)) \\ _Colin Barker_, Jul 13 2019
%o A309118 (Magma) I:=[6,6,12,18,24,24]; [n le 6 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..60]]; // _Vincenzo Librandi_, Jul 16 2019
%Y A309118 Cf. A008588, A008594.
%Y A309118 Cf. A242128 (5-fold, Star), A242129 (5-fold, Sun), A242888 (7-fold, Star), A242889 (7-fold, Sun), A242890 (8-fold, Star), A242891 (8-fold, Sun), A242892 (9-fold, Star), A242893 (9-fold, Sun), A242894 (Kite and dart, Star), A242895 (Kite and dart, Sun).
%K A309118 nonn,easy
%O A309118 1,1
%A A309118 _Felix Fröhlich_, Jul 13 2019