[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
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