# [seqfan] Tabl formula?

Ron Hardin rhhardin at att.net
Fri Apr 1 13:14:32 CEST 2011

```Is there a better formula formulation?

Row n is a polynomial of degree n, with enough initial terms from the formula
below to determine the polynomial for each entire row; but the formula itself
explicitly gives only the initial part of each row.

T(n,k)=Number of nXk binary arrays without the pattern 0 1 diagonally,
vertically, antidiagonally or horizontally

Table starts
..2..3..4..5...6...7...8...9...10...11...12....13....14....15....16.....17
..3..5..8.12..17..23..30..38...47...57...68....80....93...107...122....138
..4..7.12.20..32..49..72.102..140..187..244...312...392...485...592....714
..5..9.16.28..48..80.129.201..303..443..630...874..1186..1578..2063...2655
..6.11.20.36..64.112.192.321..522..825.1268..1898..2772..3958..5536...7599
..7.13.24.44..80.144.256.448..769.1291.2116..3384..5282..8054.12012..17548
..8.15.28.52..96.176.320.576.1024.1793.3084..5200..8584.13866.21920..33932
..9.17.32.60.112.208.384.704.1280.2304.4097..7181.12381.20965.34831..56751
.10.19.36.68.128.240.448.832.1536.2816.5120..9217.16398.28779.49744..84575
.11.21.40.76.144.272.512.960.1792.3328.6144.11264.20481.36879.65658.115402

Empirical: T(n,k) = (n+1)*2^(k-1) + (1-k)*2^(k-2) for k<n+3, and then the entire
row n is a polynomial of degree n in k.

rhhardin at mindspring.com
rhhardin at att.net (either)

```