# [seqfan] Re: Number of points with maximum norm n in A2, A3, A4, A5, D5

Richard Mathar mathar at strw.leidenuniv.nl
Mon Feb 15 14:27:38 CET 2010

To put some structure to the numbers proposed in
http://list.seqfan.eu/pipermail/seqfan/2010-February/003582.html :

Write down the central (2n+1)-nomial sequences along columns in a table A(k,n):
(optionally add a first row with all-1):
1,   3,    5,     7,      9,     11,     13,     15,     17,     19,     21,
1,   7,   19,    37,     61,     91,    127,    169,    217,    271,    331,
1,  19,   85,   231,    489,    891,   1469,   2255,   3281,   4579,   6181,
1,  51,  381,  1451,   3951,   8801,  17151,  30381,  50101,  78151, 116601,
1, 141, 1751,  9331,  32661,  88913, 204763, 418503, 782153,1363573,2248575,
1, 393, 8135, 60691, 273127, 908755,2473325,5832765,12354469,24072133,43874139,

This contains columns with centered 3-nomial A002426, centered 5-nomial A005191,
centered 7-nomial A025012, 9-nomial A025104, etc.
Then compute first differences along each of the lines, A(k,n)-A(k,n-1), to get
the number of lattice points in the A_k lattice with maximum norm n:

1,   2,    2,     2,      2,      2,      2,      2,      2,      2,      2,
1,   6,   12,    18,     24,     30,     36,     42,     48,     54,     60,
1,  18,   66,   146,    258,    402,    578,    786,   1026,   1298,   1602,
1,  50,  330,  1070,   2500,   4850,   8350,  13230,  19720,  28050,  38450,
1, 140, 1610,  7580,  23330,  56252, 115850, 213740, 363650, 581420, 885002,
1, 392, 7742, 52556, 212436, 635628,1564570,3359440,6521704,11717664,19802006,

Richard Mathar www.strw.leidenuniv.nl/~mathar