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
Examples
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, ...
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.
Crossrefs
Programs
-
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 *)
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.
Extensions
Offset and typos corrected by Alois P. Heinz, Apr 21 2012