A144965 -id:A144965 - cp's OEIS frontend

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

1, 40, 272, 888, 2080, 4040, 6960, 11032, 16448, 23400, 32080, 42680, 55392, 70408, 87920, 108120, 131200, 157352, 186768, 219640, 256160, 296520, 340912, 389528, 442560, 500200, 562640, 630072, 702688, 780680, 864240, 953560, 1048832, 1150248, 1258000, 1372280

Offset: 0

N. J. A. Sloane, Apr 15 2008

This lattice consists of all points (w,x,y,z) where w,x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Feb 05 2010

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.
Index entries for sequences related to D_4 lattice.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A035878, A105374, A110907, A144965, A175110.

I:=[1, 40, 272, 888, 2080]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012

CoefficientList[Series[(1+36*x+118*x^2+36*x^3+x^4)/(1-x)^4,{x,0,40}],x]  (* Vincenzo Librandi, Jun 27 2012 *)

From R. J. Mathar, Feb 03 2010, Feb 13 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;
a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)
G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - Colin Barker, May 24 2012
E.g.f.: 1 + 8*x*(1 + 2*x)*(5 + 2*x)*exp(x). - Elmo R. Oliveira, Aug 18 2025

a(2) corrected and sequence extended by R. J. Mathar, Feb 03 2010, Feb 13 2010

A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.

Original entry on oeis.org

1, 31, 177, 535, 1201, 2271, 3841, 6007, 8865, 12511, 17041, 22551, 29137, 36895, 45921, 56311, 68161, 81567, 96625, 113431, 132081, 152671, 175297, 200055, 227041, 256351, 288081, 322327, 359185, 398751, 441121, 486391, 534657, 586015, 640561, 698391

Offset: 0

Vincenzo Librandi, Apr 21 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, Contemp. Math. 374 (2005), 15-36, page 19.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A144965, A158187, A272039.

Magma
```
[16*n^3+10*n^2+4*n+1: n in [0..50]];
```

A272130:=n->16*n^3+10*n^2+4*n+1: seq(A272130(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2016

LinearRecurrence[{4,-6,4,-1},{1,31,177,535},50]
CoefficientList[Series[(1 + 27*x + 59*x^2 + 9*x^3)/(1 - x)^4, {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 22 2016 *)

vector(100, n, n--; 16*n^3+10*n^2+4*n+1) \\ Altug Alkan, Apr 22 2016

O.g.f.: (1+27*x+59*x^2+9*x^3)/(1-x)^4.
E.g.f.: (1+30*x+58*x^2+16*x^3)*exp(x).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
a(n) = A158187(n) + A144965(n).

cp's OEIS Frontend

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

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A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.

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cp's OEIS Frontend

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A272130 a(n) = 16*n^3 + 10*n^2 + 4*n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.