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A005902
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Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.
(Formerly M4898)
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86
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1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739, 28741, 33153, 37995, 43287, 49049, 55301, 62063, 69355, 77197, 85609, 94611, 104223, 114465, 125357, 136919, 149171, 162133, 175825, 190267, 205479
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OFFSET
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0,2
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COMMENTS
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Called "magic numbers" in some chemical contexts.
Partial sums of A005901(n). - Lekraj Beedassy, Oct 30 2003
Equals binomial transform of [1, 12, 30, 20, 0, 0, 0, ...]. - Gary W. Adamson, Aug 01 2008
Crystal ball sequence for A_3 lattice. - Michael Somos, Jun 03 2012
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REFERENCES
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H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
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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T. D. Noe, Table of n, a(n) for n = 0..1000
S. Bjornholm, Clusters, condensed matter in embryonic form, Contemp. Phys. 31 1990 pp. 309-324.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Nicolas Gastineau, Olivier Togni, Coloring of the d-th power of the face-centered cubic grid, arXiv:1806.08136 [cs.DM], 2018.
D. R. Herrick, Home Page (displays these numbers as sizes of clusters in chemistry)
Xiaogang Liang, Ilyar Hamid, and Haiming Duan, Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals,>, AIP Advances 6, 065017 (2016).
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
K. Urner, Cuboctahedral Sphere Packing
Index entries for crystal ball sequences
Index entries for sequences related to f.c.c. lattice
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = (2*n+1)*(5*n^2+5*n+3)/3.
For n > 0, n*a(n) = (Sum_{i=0..n-1} a(i)) + 2*A005891(n)*A000217(n). - Bruno Berselli, Feb 02 2011
a(-1 - n) = -a(n). - Michael Somos, Jun 03 2012
From Indranil Ghosh, Apr 08 2017: (Start)
G.f.: (x^3 + 9x^2 + 9x + 1)/(x - 1)^4.
E.g.f.: (1/3)*exp(x)*(10x^3 + 45x^2 + 36x + 3).
(End)
a(n) = A100171(n+1) - A008778(n-1) = A100174(n+1) - A000290(n) = A005917(n+1) - A006331(n) = A051673(n+1) + A000578(n). - Bruce J. Nicholson, Jul 05 2018
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EXAMPLE
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a(4) = 147 = (1, 3, 3, 1) dot (1, 12, 30, 20) = (1 + 36 + 90 + 20). - Gary W. Adamson, Aug 01 2008
G.f. = 1 + 13*x + 55*x^2 + 147*x^3 + 309*x^4 + 561*x^5 + 923*x^6 + 1415*x^7 + ...
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MAPLE
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A005902 := n -> (2*n+1)*(5*n^2+5*n+3)/3;
A005902:=(z+1)*(z**2+8*z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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f[n_] := (2n + 1)(5n^2 + 5n + 3)/3; Array[f, 36, 0] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 13, 55, 147}, 50] (* Harvey P. Dale, Oct 08 2015 *)
CoefficientList[Series[(x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4, {x, 0, 50}], x] (* Indranil Ghosh, Apr 08 2017 *)
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PROG
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(PARI) {a(n) = (2*n + 1) * (5*n^2 + 5*n + 3) / 3}; /* Michael Somos, Jun 03 2012 */
(PARI) x='x+O('x^50); Vec((x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4) \\ Indranil Ghosh, Apr 08 2017
(MAGMA) [(2*n+1)*(5*n^2+5*n+3)/3: n in [0..30]]; // G. C. Greubel, Dec 01 2017
(Python)
def a(n): return (2*n+1)*(5*n**2+5*n+3)//3
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 13 2021
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CROSSREFS
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(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
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.
Cf. A100171, A100174, A051673.
Sequence in context: A029531 A158485 A274973 * A051798 A206372 A290396
Adjacent sequences: A005899 A005900 A005901 * A005903 A005904 A005905
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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