[seqfan] Constant row and column recurrence

Ron Hardin rhhardin at att.net
Tue Dec 10 14:52:26 CET 2013


This problem produces rows and columns all satisfying the same linear recurrence.

/tmp/dsz
T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having the sum of the absolute values of all six edge and diagonal differences equal to 6

Table starts
....20....46...104...244....560..1336..3104..7504.17600..42976.101504.249664
....46....88...170...358....754..1690..3746..8722.19906..47458.110210.266818
...104...170...292...560...1100..2324..4924.10988.24284..56060.127132.300380
...244...358...560...988...1816..3616..7304.15544.33064..73288.161000.......
...560...754..1100..1816...3188..6076.11876.24340.50180.107044..............
..1336..1690..2324..3616...6076.11140.21164.42076.84556.....................
..3104..3746..4924..7304..11876.21164.39508.77060...........................
..7504..8722.10988.15544..24340.42076.77060.................................
.17600.19906.24284.33064..50180.84556.......................................
.42976.47458.56060.73288.107044.............................................

Table divided by two
....10....23....52...122...280...668..1552..3752..8800.21488.50752.124832
....23....44....85...179...377...845..1873..4361..9953.23729.55105.133409
....52....85...146...280...550..1162..2462..5494.12142.28030.63566.150190
...122...179...280...494...908..1808..3652..7772.16532.36644.80500.......
...280...377...550...908..1594..3038..5938.12170.25090.53522.............
...668...845..1162..1808..3038..5570.10582.21038.42278...................
..1552..1873..2462..3652..5938.10582.19754.38530.........................
..3752..4361..5494..7772.12170.21038.38530...............................
..8800..9953.12142.16532.25090.42278.....................................
.21488.23729.28030.36644.53522...........................................

Empirical for column k (k=2 recurrence works for k=1 as well):
k=1: a(n)=2*a(n-1)+6*a(n-2)-12*a(n-3)
k=2: a(n)=3*a(n-1)+6*a(n-2)-24*a(n-3)+4*a(n-4)+36*a(n-5)-24*a(n-6)
k=3: a(n)=3*a(n-1)+6*a(n-2)-24*a(n-3)+4*a(n-4)+36*a(n-5)-24*a(n-6)
k=4: a(n)=3*a(n-1)+6*a(n-2)-24*a(n-3)+4*a(n-4)+36*a(n-5)-24*a(n-6)
k=5: a(n)=3*a(n-1)+6*a(n-2)-24*a(n-3)+4*a(n-4)+36*a(n-5)-24*a(n-6)
k=6: a(n)=3*a(n-1)+6*a(n-2)-24*a(n-3)+4*a(n-4)+36*a(n-5)-24*a(n-6)
k=7: a(n)=3*a(n-1)+6*a(n-2)-24*a(n-3)+4*a(n-4)+36*a(n-5)-24*a(n-6)

All.solutions.for.n=k=1..
..0..2....2..0....0..0....0..2....2..2....1..2....2..0....2..0....2..1....2..1..
..2..2....0..0....2..0....1..1....0..2....0..1....2..2....1..1....1..0....0..1..
..
..1..0....1..0....1..1....0..2....0..1....1..2....1..1....0..0....2..2....0..1..
..1..2....2..1....2..0....0..0....1..2....1..0....0..2....0..2....2..0....2..1..
..

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


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