# [seqfan] Re: Constant row and column recurrence

Ron Hardin rhhardin at att.net
Tue Dec 10 16:43:32 CET 2013

```This appears to be a more general situation.  Substituting 0..3 and equal to 9 for 0..2 and equal to 6

/tmp/dta
T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X2 subblock having the sum of the absolute values of all six edge and diagonal differences equal to 9

Table starts
.....32.....80....192....512...1280...3584...9216..26624..69632.204800.540672
.....80....152....296....680...1544...4040...9992..28040..72200.209672.549896
....192....296....488....968...1992...4808..11336..30536..76872.218696.567368
....512....680....968...1640...2984...6440..14120..35624..86312.236840.602408
...1280...1544...1992...2984...4904...9512..19368..45224.104360.271784.......
...3584...4040...4808...6440...9512..16424..30632..65192.141224..............
...9216...9992..11336..14120..19368..30632..53288.104744.....................
..26624..28040..30536..35624..45224..65192.104744............................
..69632..72200..76872..86312.104360.141224...................................
.204800.209672.218696.236840.271784..........................................

Table divided by eight
.....4....10....24....64...160...448..1152..3328..8704.25600.67584.200704
....10....19....37....85...193...505..1249..3505..9025.26209.68737.202945
....24....37....61...121...249...601..1417..3817..9609.27337.70921.207241
....64....85...121...205...373...805..1765..4453.10789.29605.75301.......
...160...193...249...373...613..1189..2421..5653.13045.33973.............
...448...505...601...805..1189..2053..3829..8149.17653...................
..1152..1249..1417..1765..2421..3829..6661.13093.........................
..3328..3505..3817..4453..5653..8149.13093...............................
..8704..9025..9609.10789.13045.17653.....................................
.25600.26209.27337.29605.33973...........................................

Empirical for column k (k=2 recurrence also works for k=1):
k=1: a(n)=2*a(n-1)+8*a(n-2)-16*a(n-3)
k=2: a(n)=3*a(n-1)+8*a(n-2)-30*a(n-3)+4*a(n-4)+48*a(n-5)-32*a(n-6)
k=3: a(n)=3*a(n-1)+8*a(n-2)-30*a(n-3)+4*a(n-4)+48*a(n-5)-32*a(n-6)
k=4: a(n)=3*a(n-1)+8*a(n-2)-30*a(n-3)+4*a(n-4)+48*a(n-5)-32*a(n-6)
k=5: a(n)=3*a(n-1)+8*a(n-2)-30*a(n-3)+4*a(n-4)+48*a(n-5)-32*a(n-6)
k=6: a(n)=3*a(n-1)+8*a(n-2)-30*a(n-3)+4*a(n-4)+48*a(n-5)-32*a(n-6)
k=7: a(n)=3*a(n-1)+8*a(n-2)-30*a(n-3)+4*a(n-4)+48*a(n-5)-32*a(n-6)

All.solutions.for.n=k=1..
..1..0....1..3....3..3....3..0....0..3....3..0....0..3....0..2....2..0....3..2..
..1..3....0..1....3..0....2..2....0..0....1..1....3..3....2..3....3..2....2..0..
..
..3..1....0..2....2..0....0..0....0..0....0..3....1..1....1..1....1..0....3..0..
..0..1....3..2....2..3....3..0....0..3....2..2....0..3....3..0....3..1....0..0..
..
..3..3....2..2....3..0....0..1....3..2....2..3....2..2....0..1....3..1....0..3..
..0..3....0..3....3..3....3..1....0..2....2..0....3..0....1..3....1..0....1..1..
..
..2..3....1..3..
..0..2....1..0..

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

>________________________________
> From: Ron Hardin <rhhardin at att.net>
>To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>Sent: Tuesday, December 10, 2013 8:52 AM
>Subject: [seqfan] Constant row and column recurrence
>
>
>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)
>
>_______________________________________________
>
>Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
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