A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 18, 12, 2, 0, 1, 50, 66, 18, 2, 0, 1, 140, 330, 146, 24, 2, 0, 1, 392, 1610, 1070, 258, 30, 2, 0, 1, 1106, 7742, 7580, 2500, 402, 36, 2, 0, 1, 3138, 37058, 52556, 23330, 4850, 578, 42, 2, 0, 1, 8952, 177186, 360402, 212436, 56252, 8350, 786

Offset: 0

Author

R. J. Mathar, Mar 02 2010

Keywords

• easy
• nonn
• tabl

Comments

The values are computed starting with an auxiliary array which places the centered trinomial numbers A002426, the centered pentanomial numbers A005191, the centered 7-nomial numbers A025012 etc. into separate columns:

.1,....1,......1,.......1,........1,........1,.........1,.........1,.........1

.1,....3,......5,.......7,........9,.......11,........13,........15,........17

.1,....7,.....19,......37,.......61,.......91,.......127,.......169,.......217

.1,...19,.....85,.....231,......489,......891,......1469,......2255,......3281

.1,...51,....381,....1451,.....3951,.....8801,.....17151,.....30381,.....50101

.1,..141,...1751,....9331,....32661,....88913,....204763,....418503,....782153

.1,..393,...8135,...60691,...273127,...908755,...2473325,...5832765,..12354469

.1,.1107,..38165,..398567,..2306025,..9377467,..30162301,..82073295,.197018321

.1,.3139,.180325,.2636263,.19610233,.97464799,.370487485,1163205475,3164588407

This is a subarray of A077042. Rows are A005408, A003215, A063496, A083669 (see A077044) etc. The array A(k,n) is the first differences along each row of this auxiliary array.

Examples

			A(k,n) starts in row k=0, column n=0 as:
1,....0,......0,.......0,........0,........0,.........0,.........0,.........0
1,....2,......2,.......2,........2,........2,.........2,.........2,.........2
1,....6,.....12,......18,.......24,.......30,........36,........42,........48
1,...18,.....66,.....146,......258,......402,.......578,.......786,......1026
1,...50,....330,....1070,.....2500,.....4850,......8350,.....13230,.....19720
1,..140,...1610,....7580,....23330,....56252,....115850,....213740,....363650
1,..392,...7742,...52556,...212436,...635628,...1564570,...3359440,...6521704
1,.1106,..37058,..360402,..1907458,..7071442,..20784834,..51910994,.114945026
1,.3138,.177186,.2455938,.16973970,.77854566,.273022686,.792717990,2001382932
		

Links

• R. J. Mathar, Point counts of D_k and some A_k and E_K integer.., arXiv:1002.3844 [math .GT], variable A_k^s(n).

Crossrefs

Cf. A008458 (row k=2), A010006 (row k=3), A110907.