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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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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 inA299266for 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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