A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066

Offset

0,2

Links

Vincenzo Librandi, 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).

Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - N. J. A. Sloane, Feb 13 2013

Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.

Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002

G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012

a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013

a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018

From Shel Kaphan, Feb 28 2023: (Start)

a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.

a(n) = A001846(n) - A001845(n).

a(n) = A008412(n)*n/4. (End)

From Amiram Eldar, Mar 12 2023: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)

E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

Maple

f := proc(d, m) local i; sum( 2^i*binomial(d, i)*binomial(m-1, i-1), i=1..min(d, m)); end; # n=dimension, m=norm

Mathematica

CoefficientList[Series[2*x*(1+x)^3/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 16, 66, 192}, 50] (* Harvey P. Dale, Dec 11 2019 *)

Prog

(PARI) a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011
(Magma) [( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 2012

Crossrefs

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.

Cf. A001845, A001846, A008412, A014820.

Column 4 of A035607, A266213, A343599.

Row 4 of A113413, A119800, A122542.

Sequence in context: A222381 A110048 A094505 * A167566 A034579 A006733

Adjacent sequences: A035595 A035596 A035597 * A035599 A035600 A035601

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved