A008340 -id:A008340 - cp's OEIS frontend

A008399 Coordination sequence for E_6 lattice.

1, 72, 1062, 6696, 26316, 77688, 189810, 405720, 785304, 1408104, 2376126, 3816648, 5885028, 8767512, 12684042, 17891064, 24684336, 33401736, 44426070, 58187880, 75168252, 95901624, 120978594, 151048728, 186823368, 229078440, 278657262, 336473352, 403513236

Offset: 0

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

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008397, A008340, A019557, A019558.

[1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // G. C. Greubel, May 29 2023

1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);

LinearRecurrence[{6,-15,20,-15,6,-1},{1,72,1062,6696,26316,77688, 189810},30] (* Harvey P. Dale, Oct 24 2022 *)

[9*n*(13*n^2+7)*(n^2+1)//5 +int(n==0) for n in range(41)] # G. C. Greubel, May 29 2023

a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.
Bacher et al. give a g.f.
G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^6 = 1 + 18*x*(4+35*x+78*x^2+35*x^3+4*x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/5)*x*(360 + 2295*x + 3105*x^2 + 1170*x^3 + 117*x^4 )*exp(x). - G. C. Greubel, May 29 2023

A001361 Number of points in interior of n-th crystal ball in E_8 lattice.

Original entry on oeis.org

1, 241, 26641, 338401, 2240161, 10242001, 36638641, 109907521, 288653761, 683158321, 1486914001, 3020862241, 5793372001, 10579330321, 18522042481, 31261968001, 51096647041, 81176500081, 125741512081, 190404140641, 282484116001, 411401129041, 589131731761

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A008340, A008349.

a:= n-> 57/7*n^8-108/7*n^7+30*n^6-72*n^5+39*n^4-36*n^3+300/7*n^2+24/7*n+1:
seq (a(n), n=1..30);

CoefficientList[Series[-(x^8 + 232 x^7 + 7228 x^6 + 55384 x^5 + 133510 x^4 + 107224 x^3 + 24508 x^2 + 232 x + 1)/(x - 1)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 13 2013 *)

G.f.: -x*(x^8 + 232*x^7 + 7228*x^6 + 55384*x^5 + 133510*x^4 + 107224*x^3 + 24508*x^2 + 232*x + 1)/(x - 1)^9. - Colin Barker, Oct 29 2012

A323282 Coordination sequence for Leech lattice.

Original entry on oeis.org

1, 196560, 39462040800

Offset: 0

N. J. A. Sloane, Jan 12 2019

Coordination sequence for graph G having a vertex for each point of the Leech lattice, with each vertex joined by an edge to its 196560 nearest neighbors.

No formula or recurrence is presently known (see page xxxvi of the sphere packing book).

			The graph G has degree 196560, so a(1)=196560.
Let L(i) denote the set of vectors in the Leech lattice having norm 2i, and let c(i) = |L(i)| = A008408(i). The first few sets L(i) are described on page 181 of the ATLAS and in Table 4.13 of the sphere packing book.
The vertices at edge-distance 2 from the vertex 0^24 consist of the sets L(3), L(4), L(5), L(6), and the doubles of L(2), so a(2) = c(2)+c(3)+...+c(6) = 39462040800.
To prove this, note that the group Aut(G) = Co_0 acts transitively on each of the sets L(2), L(3), L(4), L(5), and L(7) (cf. A323273). It is easy to find one representative from each set L(3), L(4), L(5) among the sums of pairs of minimal vectors. There are two orbits on L(6) and again it is easy to find a representative from each orbit, for example
[3, 3, 3, 3, 3, 3, 3, 3, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]
and
[2, 2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2].
Among the sums of pairs of minimal vectors there are no vectors from L(7), so there is no contribution from c(7).
Finally, the only possibility for a vector of norm 16 is the double of a minimal vector, and there are c(2) = 196560 of these, all in one obit under the group.
So a(2) = c(3)+c(4)+c(5)+c(6)+c(2) = 39462040800.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. 181.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993; see page xxxvi and Tables 4.13 and 4.14.

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

Cf. A007900, A008340, A008408, A323273.

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A008399 Coordination sequence for E_6 lattice.

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A001361 Number of points in interior of n-th crystal ball in E_8 lattice.

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A323282 Coordination sequence for Leech lattice.

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