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A007202
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Crystal ball sequence for hexagonal close-packing.
(Formerly M4899)
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55
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1, 13, 57, 153, 323, 587, 967, 1483, 2157, 3009, 4061, 5333, 6847, 8623, 10683, 13047, 15737, 18773, 22177, 25969, 30171, 34803, 39887, 45443, 51493, 58057, 65157, 72813, 81047, 89879, 99331, 109423, 120177, 131613, 143753, 156617
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
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FORMULA
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Nearest integer to (7/8)*( (n+1)^4 - n^4 ).
G.f.: (x^4+10*x^3+20*x^2+10*x+1)/(x-1)^4/(x+1).
a(n) = 7*(2*n+1)*(2*n^2+2*n+1)/8 +(-1)^n/8. - R. J. Mathar, Mar 24 2011
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
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
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MATHEMATICA
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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 *)
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PROG
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(PARI) j=[]; for(n=0, 75, j=concat(j, round((7/8)*((n+1)^4-n^4)))); j
(Python)
def a(n): return round((7/8)*((n+1)**4-n**4))
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CROSSREFS
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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.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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