A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

4, 8, 6, 12, 18, 8, 16, 38, 32, 10, 20, 66, 88, 50, 12, 24, 102, 192, 170, 72, 14, 28, 146, 360, 450, 292, 98, 16, 32, 198, 608, 1002, 912, 462, 128, 18, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 40, 326, 1408, 3530, 5336, 4942, 2816, 978, 200, 22

Offset

1,1

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.

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

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

Formula

A(m,n) = A(m,n-1) + A(m-1,n) + A(m-1,n-1), A(m,0)=1, A(0,0)=1, A(0,n)=2.

Example

The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron.
The third column of the table is: 12, 38, 88, 170, 292, 462, 688, 978, 1340, ... = A035597 = number of points of L1 norm 3 in cubic lattice Z^n.
The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
The main diagonal seems to be A050146.
Square array A(n,k) begins:
   4,   8,   12,   16,    20,    24,     28,     32,      36, ...
   6,  18,   38,   66,   102,   146,    198,    258,     326, ...
   8,  32,   88,  192,   360,   608,    952,   1408,    1992, ...
  10,  50,  170,  450,  1002,  1970,   3530,   5890,    9290, ...
  12,  72,  292,  912,  2364,  5336,  10836,  20256,   35436, ...
  14,  98,  462, 1666,  4942, 12642,  28814,  59906,  115598, ...
  16, 128,  688, 2816,  9424, 27008,  68464, 157184,  332688, ...
  18, 162,  978, 4482, 16722, 53154, 148626, 374274,  864146, ...
  20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...

Maple

A:= proc(m, n)  option remember;
      `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Apr 21 2012

Mathematica

A[m_, n_] := A[m, n] = If[n == 0, 1, If[m == 0, 2, A[m, n-1] + A[m-1, n] + A[m-1, n-1]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Prog

Excel cell formula: =Z(-1)S(-1)+Z(-1)S+ZS(-1). The very first row (not included into the table) contains the initialization values: 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... The very first column (not included into the table) contains the initialization values: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... Note that the first cell is common to both the initialization row and initialization column and it equal to 1.

Crossrefs

Other versions: A035607, A113413, A122542, A266213.

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.

Sequence in context: A288189 A335159 A064494 * A330685 A063723 A323057

Adjacent sequences: A119797 A119798 A119799 * A119801 A119802 A119803

Keyword

easy,nonn,tabl

Author

Thomas Wieder, Jul 30 2006, Aug 06 2006

Extensions

Offset and typos corrected by Alois P. Heinz, Apr 21 2012

Status

approved