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

0,2

Comments

First 20 terms computed by Davide M. Proserpio using ToposPro.

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).

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

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 *)

Prog

(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
(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

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.

Sequence in context: A110200 A063404 A102177 * A219521 A215178 A058893

Adjacent sequences: A299263 A299264 A299265 * A299267 A299268 A299269

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 07 2018

Extensions

a(21)-a(40) from Davide M. Proserpio, Feb 12 2018

Status

approved