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A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals). 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

REFERENCES

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

J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.

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.

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.

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-Fran├žois 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: A005531 A288189 A064494 * A063723 A028269 A019650

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

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Last modified September 20 16:14 EDT 2017. Contains 292276 sequences.