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Coordination sequence for "tea" 3D uniform tiling.
51

%I #25 Jun 12 2024 17:15:16

%S 1,10,33,73,128,199,285,388,506,640,789,955,1136,1333,1545,1774,2018,

%T 2278,2553,2845,3152,3475,3813,4168,4538,4924,5325,5743,6176,6625,

%U 7089,7570,8066,8578,9105,9649,10208,10783,11373,11980,12602,13240,13893

%N Coordination sequence for "tea" 3D uniform tiling.

%C First 20 terms computed by _Davide M. Proserpio_ using ToposPro.

%H B. Grünbaum, <a href="https://faculty.washington.edu/moishe/branko/BG199.Uniform%20Tilings%20of%203-Space.pdf">Uniform tilings of 3-space</a>, Geombinatorics, 4 (1994), 49-56. See tiling #4.

%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/tea">The tea tiling (or net)</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).

%F From _Colin Barker_, Feb 11 2018: (Start)

%F G.f.: (1 + 8*x + 14*x^2 + 17*x^3 + 14*x^4 + 8*x^5 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6. (End)

%F [I suspect Barker's formulas only conjectures. - _N. J. A. Sloane_, Jun 12 2024]

%F If the above formulas are true, then a(n) = (31 - 3*(-1)^n + 126*n^2 + 4*A056594(n))/16 for n > 0. - _Stefano Spezia_, Jun 08 2024

%t LinearRecurrence[{2,-1,0,1,-2,1},{1,10,33,73,128,199,285},50] (* _Harvey P. Dale_, May 09 2022 *)

%o (PARI) a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,-2,1,0,-1,2]^n*[1;10;33;73;128;199])[1,1] \\ _Charles R Greathouse IV_, Oct 18 2022

%Y See A299286 for partial sums.

%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.

%Y Cf. A056594.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 10 2018