A299266 Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.
1, 5, 9, 22, 37, 57, 82, 117, 145, 178, 229, 281, 322, 377, 445, 514, 577, 645, 730, 825, 901, 982, 1093, 1205, 1294, 1397, 1525, 1654, 1765, 1881, 2026, 2181, 2305, 2434, 2605, 2777, 2914, 3065, 3253, 3442, 3601, 3765, 3970, 4185, 4357, 4534, 4765, 4997, 5182, 5381, 5629, 5878, 6085, 6297, 6562, 6837
Offset: 0
References
- B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #8.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, Applied Topological Analysis of Crystal Structures with the Program Package ToposPro, Cryst. Growth Des. 2014, 14, 3576-3586.
- Reticular Chemistry Structure Resource (RCSR), The cab tiling (or net)
- Davide M. Proserpio, Summary of the 28 uniform 3D tilings and their coordination sequences (produced by ToposPro)
- Index entries for linear recurrences with constant coefficients, signature (1,-1,2,0,0,0,-2,1,-1,1).
Crossrefs
See A299267 for partial sums.
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.
Programs
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Magma
I:=[22, 37, 57, 82, 117, 145, 178,229, 281,322]; [1,5,9] cat [n le 10 select I[n] else Self(n-1) -Self(n-2) +2*Self(n-3)-2*Self(n-7)+Self(n-8)-Self(n-9) + Self(n-10): n in [1..30]]; // G. C. Greubel, Feb 20 2018
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Mathematica
CoefficientList[Series[(4*x^12-4*x^11+x^10+5*x^8+20*x^7+18*x^6+24*x^5 +14*x^4+16*x^3+5*x^2+4*x+1)/((1-x)*(1-x^2)*(1-x^3)*(1+x^2)^2), {x,0, 50}], x] (* G. C. Greubel, Feb 20 2018 *) -
PARI
Vec((1 + 4*x + 5*x^2 + 16*x^3 + 14*x^4 + 24*x^5 + 18*x^6 + 20*x^7 + 5*x^8 + x^10 - 4*x^11 + 4*x^12) / ((1 - x)^3*(1 + x)*(1 + x^2)^2*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 15 2018
Formula
G.f.: (4*x^12 -4*x^11 +x^10 +5*x^8 +20*x^7 +18*x^6 +24*x^5 +14*x^4 +16*x^3 +5*x^2 +4*x +1)/((1-x)*(1-x^2)*(1-x^3)*(1+x^2)^2). - N. J. A. Sloane, Feb 12 2018
a(n) = a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-10) for n>12. - Colin Barker, Feb 15 2018
Extensions
a(21)-a(40) from Davide M. Proserpio, Feb 12 2018
Comments