A007899 Coordination sequence for hexagonal close-packing.
1, 12, 44, 96, 170, 264, 380, 516, 674, 852, 1052, 1272, 1514, 1776, 2060, 2364, 2690, 3036, 3404, 3792, 4202, 4632, 5084, 5556, 6050, 6564, 7100, 7656, 8234, 8832, 9452, 10092, 10754, 11436, 12140, 12864, 13610, 14376, 15164, 15972, 16802, 17652, 18524, 19416, 20330, 21264, 22220
Offset: 0
References
- B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #2.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
- Reticular Chemistry Structure Resource (RCSR), The hcp tiling (or net)
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Crossrefs
For partial sums see A007202.
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:=[1,12,44,96,170]; [n le 5 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 16 2014
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Magma
[1] cat [2 + Floor(21*n^2/2): n in [1..50]]; // G. C. Greubel, Feb 20 2018
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Mathematica
Join[{1},Floor[(21Range[40]^2)/2]+2] (* or *) Join[{1},LinearRecurrence[ {2,0,-2,1},{12,44,96,170},40]] (* Harvey P. Dale, Feb 15 2014 *) CoefficientList[Series[(x^4 + 10 x^3 + 20 x^2 + 10 x + 1)/(1 - x)^3/(x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 16 2014 *) -
PARI
for(n=0,50, print1(if(n==0,1, 2 + floor(21*n^2/2)), ", ")) \\ G. C. Greubel, Feb 20 2018
Formula
a(n) = floor( 21*n^2 / 2 ) + 2, for n>= 1.
G.f.: (x^4 +10*x^3 +20*x^2 +10*x +1)/((1+x)*(1-x)^3).
a(0)=1, a(1)=12, a(2)=44, a(3)=96, a(4)=170, a(n)=2*a(n-1)-2*a(n-3)+ a(n-4). - Harvey P. Dale, Feb 15 2014
a(n) = (21/2)*n^2 + 7/4 + (1/4)*(-1)^n - 0^n. - Eric Simon Jacob, Feb 12 2023
E.g.f.: ((4 + 21*x + 21*x^2)*cosh(x) + 3*(1 + 7*x + 7*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Mar 14 2024