A007202 Crystal ball sequence for hexagonal close-packing. (Formerly M4899)

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

Offset

0,2

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Xiaogang Liang, Ilyar Hamid, and Haiming Duan, Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals, AIP Advances 6, 065017 (2016).

Index entries for crystal ball sequences

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

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

Mathematica

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 *)

Prog

(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))
print([a(n) for n in range(36)]) # Michael S. Branicky, Jan 13 2021

Crossrefs

Partial sums of 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.

Sequence in context: A212053 A264853 A210290 * A368813 A222161 A218593

Adjacent sequences: A007199 A007200 A007201 * A007203 A007204 A007205

Keyword

nonn,easy,nice

Author

N. J. A. Sloane and J. H. Conway

Extensions

More terms from Jason Earls, Jul 14 2001

Status

approved