%I M4899 #57 Mar 14 2024 23:25:59

%S 1,13,57,153,323,587,967,1483,2157,3009,4061,5333,6847,8623,10683,

%T 13047,15737,18773,22177,25969,30171,34803,39887,45443,51493,58057,

%U 65157,72813,81047,89879,99331,109423,120177,131613,143753,156617

%N Crystal ball sequence for hexagonal close-packing.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007202/b007202.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 Xiaogang Liang, Ilyar Hamid, and Haiming Duan, <a href="https://doi.org/10.1063/1.4954741">Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals</a>, AIP Advances 6, 065017 (2016).

%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).

%F Nearest integer to (7/8)*( (n+1)^4 - n^4 ).

%F G.f.: (x^4+10*x^3+20*x^2+10*x+1)/(x-1)^4/(x+1).

%F a(n) = 7*(2*n+1)*(2*n^2+2*n+1)/8 +(-1)^n/8. - _R. J. Mathar_, Mar 24 2011

%F a(0)=1, a(1)=13, a(2)=57, a(3)=153, a(4)=323, a(n)=3*a(n-1)- 2*a(n-2)- 2*a(n-3)+3*a(n-4)-a(n-5). - _Harvey P. Dale_, Jul 15 2011

%F E.g.f.: ((4 + 49*x + 63*x^2 + 14*x^3)*cosh(x) + (3 + 49*x + 63*x^2+ 14*x^3)*sinh(x))/4. - _Stefano Spezia_, Mar 14 2024

%t Table[Floor[(7((n+1)^4-n^4)+4)/8],{n,0,40}] (* or *) LinearRecurrence[ {3,-2,-2,3,-1},{1,13,57,153,323},40] (* _Harvey P. Dale_, Jul 15 2011 *)

%o (PARI) j=[]; for(n=0,75,j=concat(j,round((7/8)*((n+1)^4-n^4)))); j

%o (Python)

%o def a(n): return round((7/8)*((n+1)**4-n**4))

%o print([a(n) for n in range(36)]) # _Michael S. Branicky_, Jan 13 2021

%Y Partial sums of A007899.

%Y 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 nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_ and _J. H. Conway_

%E More terms from _Jason Earls_, Jul 14 2001