[seqfan] Re: Table with Small Recurrences

[Ron Hardin]

A few more columns make the pattern clearer
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
k=8: a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4)-a(n-5)+a(n-6)-a(n-7)+a(n-8) for n>14
k=9: a(n)=2*a(n-2)+a(n-10) for n>17
k=10: 
a(n)=a(n-1)+a(n-2)-a(n-3)+a(n-4)-a(n-5)+a(n-6)-a(n-7)+a(n-8)-a(n-9)+a(n-10) for 
n>18
k=11: a(n)=2*a(n-2)+a(n-12) for n>21

so:
k odd  a(n)=2*a(n-2)+a(n-k-1) for n>2k-1
k even a(n)=a(n-1)+sum{i in 2..k}(-1^i*a(n-i)) for n>2k-2

if I've abstracted right.


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



----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: seqfan at list.seqfan.eu
> Sent: Sun, January 22, 2012 5:58:18 PM
> Subject: [seqfan] Table with Small Recurrences
> 
> 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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