A008530 Coordination sequence for 4-dimensional primitive di-isohexagonal orthogonal lattice.

1, 12, 60, 180, 408, 780, 1332, 2100, 3120, 4428, 6060, 8052, 10440, 13260, 16548, 20340, 24672, 29580, 35100, 41268, 48120, 55692, 64020, 73140, 83088, 93900, 105612, 118260, 131880, 146508, 162180, 178932, 196800, 215820, 236028, 257460, 280152, 304140, 329460, 356148, 384240

Offset

0,2

Comments

Coordination sequence for 4-dimensional cyclotomic lattice Z[zeta_12].

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.

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 (4,-6,4,-1).

Formula

G.f.: (1+4*x+x^2)^2/(1-x)^4. - Colin Barker, Apr 14 2012

3*a(n) = (2*n+1)^3 + (2*n-1)^3 + (n+1)^3 + (n-1)^3 for n>0. - Bruno Berselli, Jan 31 2013

E.g.f.: 1 + x*(12 + 18*x + 6*x^2)*exp(x). - G. C. Greubel, Nov 10 2019

Example

3*a(5) = 2340 = (2*5+1)^3 + (2*5-1)^3 + (5+1)^3 + (5-1)^3. - Bruno Berselli, Jan 31 2013

Maple

1, seq( 6*k^3+6*k, k=1..45);

Mathematica

CoefficientList[Series[(1+4*x+x^2)^2/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Apr 16 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 12, 60, 180, 408}, 45] (* G. C. Greubel, Nov 10 2019 *)

Prog

(Magma) [1]cat[6*n^3+6*n: n in [1..45]]; // Vincenzo Librandi, Apr 16 2012
(PARI) vector(46, n, if(n==1, 1, 6*(n-1)*(1+(n-1)^2)) ) \\ G. C. Greubel, Nov 10 2019
(Sage) [1]+[6*n*(1+n^2) for n in (1..45)] # G. C. Greubel, Nov 10 2019
(GAP) Concatenation([1], List([1..45], n-> 6*n*(1+n^2) )); # G. C. Greubel, Nov 10 2019

Crossrefs

Sequence in context: A300758 A332544 A279509 * A033486 A112415 A174642

Adjacent sequences: A008527 A008528 A008529 * A008531 A008532 A008533

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved