|
| |
|
|
A299272
|
|
Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes.
|
|
51
|
|
|
|
1, 6, 18, 37, 63, 99, 142, 189, 249, 317, 384, 468, 562, 648, 756, 877, 981, 1113, 1262, 1383, 1539, 1717, 1854, 2034, 2242, 2394, 2598, 2837, 3003, 3231, 3502, 3681, 3933, 4237, 4428, 4704, 5042, 5244, 5544, 5917, 6129, 6453, 6862, 7083, 7431, 7877, 8106, 8478, 8962, 9198
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
First 20 terms computed by Davide M. Proserpio using ToposPro.
The tiling is called "3-RCO-trille" in Conway, Burgiel, Goodman-Strauss, 2008, p. 297. - Felix Fröhlich, Feb 11 2018
|
|
|
REFERENCES
|
J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #5.
|
|
|
LINKS
|
G. C. Greubel, Table of n, a(n) for n = 0..5000
Reticular Chemistry Structure Resource (RCSR), The flu tiling (or net)
Wikipedia, Tetrahedral-octahedral honeycomb - Runcic cubic honeycomb
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).
|
|
|
FORMULA
|
Conjectures from Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + x)^3*(1 + x^2)*(1 + 3*x + 5*x^2 + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>9.
(End)
G.f.: (x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3 / (1-x^3)^3. - N. J. A. Sloane, Feb 12 2018 (This confirms my conjecture from Feb 10 2018 and the above conjecture from Colin Barker.)
a(n) = (60 + 104*n^2 + (n^2 - 6)*cos(2*n*Pi/3) - 3*sqrt(3)*n*sin(2*n*Pi/3))/27 for n > 0. - Stefano Spezia, Jan 23 2022
|
|
|
MATHEMATICA
|
CoefficientList[Series[(x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3, {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
|
|
|
PROG
|
(PARI) x='x+O('x^30); Vec((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3) \\ G. C. Greubel, Feb 20 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3)) // G. C. Greubel, Feb 20 2018
|
|
|
CROSSREFS
|
See A299273 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: A180438 A202366 A185223 * A101853 A132432 A005899
Adjacent sequences: A299269 A299270 A299271 * A299273 A299274 A299275
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane, Feb 10 2018
|
|
|
EXTENSIONS
|
a(21)-a(40) from Davide M. Proserpio, Feb 12 2018
|
|
|
STATUS
|
approved
|
| |
|
|
