[seqfan] Table with Small Recurrences

Ron Hardin rhhardin at att.net
Sun Jan 22 23:58:18 CET 2012

```A current table that's finishing up turns up some surprisingly small recurrences

T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with the number of clockwise edge
increases in every 2X2 subblock differing from each horizontal or vertical
neighbor

Table starts
..16..24..48...72..144..216..432..648..1296..1944..3888..5832.11664.17496.34992
..24..40..64..104..168..272..440..712..1152..1864..3016..4880..7896.12776.20672
..48..64.124..160..292..384..708..928..1708..2240..4124..5408..9956.13056.24036
..72.104.160..256..384..576..864.1312..1984..3008..4544..6880.10400.15744.23808
.144.168.292..384..736..896.1568.1920..3392..4224..7520..9344.16608.20608.36608
.216.272.384..576..896.1408.2048.2944..4224..6144..8960.13184.19328.28416.41600
.432.440.708..864.1568.2048.3904.4608..7872..9216.15808.18944.32960.39936.69824
.648.712.928.1312.1920.2944.4608.7168.10240.14336.19968.28160.39936.57344.82432

Some solutions for n=4 k=3
..1..0..0..1....0..1..1..0....1..1..1..0....0..0..1..1....1..1..0..0..
..0..1..1..0....0..1..1..1....0..1..1..0....0..0..1..1....1..1..0..0..
..1..1..1..0....1..0..1..1....1..0..0..1....1..1..0..0....0..0..1..1..
..1..1..0..1....1..1..0..1....0..0..0..1....1..1..0..0....0..0..1..1..
..1..0..1..1....1..1..1..0....0..0..1..0....0..0..1..0....1..1..0..1..

Empirical for column k:
k=1: a(n)=3*a(n-2)
k=2: a(n)=a(n-1)+a(n-2)
k=3: a(n)=2*a(n-2)+a(n-4) for n>5
k=4: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4) for n>6
k=5: a(n)=2*a(n-2)+a(n-6) for n>9
k=6: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4)-a(n-5)+a(n-6) for n>10
k=7: a(n)=2*a(n-2)+a(n-8) for n>13

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

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