OFFSET
0,2
COMMENTS
First 20 terms computed byDavide 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)
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).
FORMULA
Conjectures fromColin 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 fromColin 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
SeeA299273 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 inA299266 for overview.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 10 2018
EXTENSIONS
a(21)-a(40) fromDavide M. Proserpio, Feb 12 2018
STATUS
approved