[seqfan] Symmetric recurrence

[Ron Hardin]

Some sudden regularities


The recurrence for column 4 is symmetric, recurrence for column k is order 2^(k-1), lead coefficient is 2^k



/tmp/dju
T(n,k)=Number of nXk 0..2 arrays with top left element 0, horizontal differences mod 3 never 1, vertical differences mod 3 never -1, and antidiagonal differences never 0

Table starts
...1.....2......4.......8........16.........32...........64...........128
...2.....5.....13......34........89........233..........610..........1597
...4....13.....44.....153.......536.......1881.........6604.........23189
...8....34....153.....711......3357......15952........75965........362012
..16....89....536....3357.....21464.....138645.......899860.......5852687
..32...233...1881...15952....138645....1220881.....10826489......96353860
..64...610...6604...75965....899860...10826489....131393852....1602580515
.128..1597..23189..362012...5852687...96353860...1602580515...26816872052
.256..4181..81428.1725628..38099072..859094433..19601243880..450388122809
.512.10946.285937.8226525.248105251.7666628193.240116025296.7581011149672

Some.solutions.for.n=4.k=4..
..0..2..2..1....0..2..2..1....0..2..1..0....0..2..1..1....0..2..1..0..
..1..0..2..2....0..0..0..2....1..0..2..1....0..2..2..1....0..2..1..1..
..2..1..0..0....1..1..1..0....1..0..2..2....0..0..0..2....0..0..2..2..
..0..2..1..1....2..2..2..1....1..1..0..2....1..1..0..0....1..0..0..2..

Empirical for column k:
k=1: a(n)=2*a(n-1)
k=2: a(n)=3*a(n-1)-a(n-2)
k=3: a(n)=5*a(n-1)-6*a(n-2)+3*a(n-3)-a(n-4)
k=4: a(n)=9*a(n-1)-28*a(n-2)+47*a(n-3)-55*a(n-4)+47*a(n-5)-28*a(n-6)+9*a(n-7)-a(n-8)
k=5: a(n)=17*a(n-1)-120*a(n-2)+505*a(n-3)-1511*a(n-4)+3524*a(n-5)-6668*a(n-6)+10300*a(n-7)-... [order 16]
k=6: a(n)=33*a(n-1)-496*a(n-2)+4675*a(n-3)-31879*a(n-4)+170672*a(n-5)-754133*a(n-6)+2833601*a(n-7)-... [order 32]
k=7: a(n)=65*a(n-1)-2016*a(n-2)+40313*a(n-3)-591045*a(n-4)+6834735*a(n-5)-65337979*a(n-6)+...[order 64]


Column 2 is A001519

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