This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007899 #54 Mar 14 2024 23:26:35
%S A007899 1,12,44,96,170,264,380,516,674,852,1052,1272,1514,1776,2060,2364,
%T A007899 2690,3036,3404,3792,4202,4632,5084,5556,6050,6564,7100,7656,8234,
%U A007899 8832,9452,10092,10754,11436,12140,12864,13610,14376,15164,15972,16802,17652,18524,19416,20330,21264,22220
%N A007899 Coordination sequence for hexagonal close-packing.
%D A007899 B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #2.
%H A007899 Vincenzo Librandi, <a href="/A007899/b007899.txt">Table of n, a(n) for n = 0..1000</a>
%H A007899 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H A007899 M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.
%H A007899 M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H A007899 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/nets/hcp">The hcp tiling (or net)</a>
%H A007899 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A007899 a(n) = floor( 21*n^2 / 2 ) + 2, for n>= 1.
%F A007899 G.f.: (x^4 +10*x^3 +20*x^2 +10*x +1)/((1+x)*(1-x)^3).
%F A007899 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
%F A007899 a(n) = (21/2)*n^2 + 7/4 + (1/4)*(-1)^n - 0^n. - _Eric Simon Jacob_, Feb 12 2023
%F A007899 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
%t A007899 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 *)
%t A007899 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 *)
%o A007899 (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
%o A007899 (PARI) for(n=0,50, print1(if(n==0,1, 2 + floor(21*n^2/2)), ", ")) \\ _G. C. Greubel_, Feb 20 2018
%o A007899 (Magma) [1] cat [2 + Floor(21*n^2/2): n in [1..50]]; // _G. C. Greubel_, Feb 20 2018
%Y A007899 For partial sums see A007202.
%Y A007899 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.
%K A007899 nonn,easy
%O A007899 0,2
%A A007899 _N. J. A. Sloane_ and _J. H. Conway_