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%I #33 Mar 14 2024 16:37:28
%S 1,6,18,40,72,112,162,220,288,364,450,544,648,760,882,1012,1152,1300,
%T 1458,1624,1800,1984,2178,2380,2592,2812,3042,3280,3528,3784,4050,
%U 4324,4608,4900,5202,5512,5832,6160,6498,6844,7200,7564,7938,8320,8712,9112,9522,9940,10368,10804,11250,11704
%N Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.6.3.6 2D tiling (cf. A008579).
%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #18.
%H Colin Barker, <a href="/A299256/b299256.txt">Table of n, a(n) for n = 0..1000</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/kag">The kag tiling (or net)</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F G.f.: (1 + 2*x)*(x^4 - 2*x^3 - 2*x^2 - 2*x - 1) / ((x - 1)^3*(x + 1)).
%F From _Colin Barker_, Feb 09 2018: (Start)
%F a(n) = 9*n^2 / 2 for n>1.
%F a(n) = (9*n^2 - 1) / 2 for n>1.
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5. (End)
%F E.g.f.: (2 + 4*x + 9*x*(x + 1)*cosh(x) + (9*x^2 + 9*x - 1)*sinh(x))/2. - _Stefano Spezia_, Mar 14 2024
%p seq(coeff(series((1+2*x)*(x^4-2*x^3-2*x^2-2*x-1)/((x-1)^3*(1+x)),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Oct 26 2018
%t Join[{1, 6}, LinearRecurrence[{2, 0, -2, 1}, {18, 40, 72, 112}, 50]] (* _Vincenzo Librandi_, Oct 26 2018 *)
%o (PARI) Vec((1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4) / ((1 - x)^3*(1 + x)) + O(x^60)) \\ _Colin Barker_, Feb 09 2018
%o (Magma) [1, 6] cat [9*n^2 div 2: n in [2..50]]; // _Vincenzo Librandi_, Oct 26 2018
%o (GAP) a:=[18,40,72,112];; for n in [5..50] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; Concatenation([1,6],a); # _Muniru A Asiru_, Oct 26 2018
%Y Cf. A008579.
%Y For partial sums see A299262.
%Y The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 07 2018