A299268 - OEIS

%I #26 Mar 14 2024 23:26:20

%S 1,6,18,48,78,126,182,240,330,390,522,576,758,798,1038,1056,1362,1350,

%T 1730,1680,2142,2046,2598,2448,3098,2886,3642,3360,4230,3870,4862,

%U 4416,5538,4998,6258,5616,7022,6270,7830,6960,8682,7686,9578,8448,10518,9246

%N Coordination sequence for "crs" 3D uniform tiling formed from tetrahedra and truncated tetrahedra.

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

%D B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #6.

%H Colin Barker, <a href="/A299268/b299268.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F G.f.: (x^6 + 27*x^4 + 30*x^3 + 15*x^2 + 6*x + 1) / (1 - x^2)^3.

%F From _Colin Barker_, Feb 09 2018: (Start)

%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.

%F a(n) = (11*n^2 - 6*n + 4) / 2 for n>0 and even.

%F a(n) = 3 * (3*n^2 + 2*n - 1) / 2 for n odd. (End)

%F E.g.f.: ((11*x^2 + 15*x + 4)*cosh(x) + (9*x^2 + 5*x - 3)*sinh(x) - 2)/2. - _Stefano Spezia_, Mar 14 2024

%t CoefficientList[Series[(x^6+27*x^4+30*x^3+15*x^2+6*x+1)/(1-x^2)^3, {x, 0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)

%o (PARI) Vec((1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^3*(1 + x)^3) + O(x^60)) \\ _Colin Barker_, Feb 09 2018

%o (Magma) I:=[18, 48, 78, 126, 182, 240, 330]; [1,6] cat [n le 6 select I[n] else 3*Self(n-2) -3*Self(n-4) + Self(n-6): n in [1..30]]; // _G. C. Greubel_, Feb 20 2018

%Y See A299269 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.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 07 2018