login
Coordination sequence for "reo-e" 3D uniform tiling.
51

%I #25 Apr 23 2023 13:06:30

%S 1,6,19,41,72,114,166,224,288,364,454,550,648,758,886,1020,1152,1296,

%T 1462,1634,1800,1978,2182,2392,2592,2804,3046,3294,3528,3774,4054,

%U 4340,4608,4888,5206,5530,5832,6146,6502,6864,7200,7548,7942,8342,8712,9094,9526,9964,10368

%N Coordination sequence for "reo-e" 3D uniform tiling.

%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 #9.

%H Colin Barker, <a href="/A299281/b299281.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).

%F G.f.: (x+1)*(x^3+x^2+1)*(x^6-2*x^5+x^4+3*x^2+2*x+1) / ((x^2+1)^2*(1-x)^3). - _N. J. A. Sloane_, Feb 12 2018

%F a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n>8. - _Colin Barker_, Feb 14 2018

%F a(n) = (9*n^2 + 4*(1 - A056594(n)) - (n - 4)*A056594(n+1))/2 for n > 3. - _Stefano Spezia_, Apr 23 2023

%o (PARI) Vec((1 + x)*(1 + x^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ _Colin Barker_, Feb 14 2018

%Y See A299282 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

%E a(21)-a(40) from _Davide M. Proserpio_, Feb 12 2018