[seqfan] Re: Long Transients before Polynomial

[Ron Hardin]

Another case https://oeis.org/A240338

/tmp/ehg
T(n,k)=Number of nXk 0..3 arrays with no element equal to two plus the sum of elements to its left or two plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4

Table starts
..2....4.......7........11...........16............22.............29
..4...15......48.......125..........284...........582...........1097
..7...48.....316......1543.........6271.........22116..........69596
.11..125....1543.....14456.......110327........716770........4106515
.16..284....6271....110327......1607848......19629542......208224462
.22..582...22116....716770.....19629542.....455506837.....9073358239
.29.1097...69596...4106515....208224462....9073358239...342013040533
.37.1932..199504..21225132...1979743527..160455447637.11361329151015
.46.3219..528924.100450928..17168302936.2579449716281...............
.56.5123.1310622.439636230.137234695613.............................

Empirical for column k:
k=1: a(n) = (1/2)*n^2 + (1/2)*n + 1
k=2: [polynomial of degree 5] for n>2
k=3: [polynomial of degree 13] for n>12
k=4: [polynomial of degree 30] for n>35
k=5: [polynomial of degree 69] for n>88

Some solutions for n=4 k=4  
..0..0..3..3....0..0..0..0....0..0..0..0....3..0..0..3....0..3..3..3..
..0..0..3..3....0..3..0..3....0..0..3..3....0..3..3..2....0..0..3..2..
..3..3..2..1....0..0..3..2....0..3..2..2....3..0..2..2....3..3..3..0..
..3..2..1..2....0..0..0..0....0..3..2..0....3..3..2..0....3..2..2..2..



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



>________________________________
> From: Ron Hardin <rhhardin at att.net>
>To: "seqfan at list.seqfan.eu" <seqfan at list.seqfan.eu> 
>Sent: Tuesday, March 25, 2014 3:46 PM
>Subject: [seqfan] Long Transients before Polynomial
> 
>
>An unusually long transient before a polynomial sets in came up (k=3,4,5 formulas).  I don't know if a reason is obvious.
>
>
>/tmp/egv
>T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or one plus the sum of the elements above it or two plus the sum of the elements diagonally to its northwest, modulo 4
>
>Table starts
>..2...3.....4........5.........6..........7..........8..........9..........10
>..3...8....17.......35........64........109........176........272.........405
>..4..17....68......244.......777.......2221.......5853......14488.......34057
>..5..35...244.....1613......9066......46260.....214126.....921674.....3745690
>..6..64...777.....9066.....94613.....874352....7359682...57010666...415293446
>..7.109..2221....46260....874352...15039319..232886648.3315673203.44101959522
>..8.176..5853...214126...7359682..232886648.6712505927.......................
>..9.272.14488...921674..57010666.3315673203..................................
>.10.405.34057..3745690.415293446.............................................
>.11.584.76495.14493362.......................................................
>
>Some.solutions.for.n=4.k=4..
>..0..0..0..0....0..0..0..3....3..3..0..0....0..0..0..0....0..0..3..3..
>..0..0..0..3....3..3..0..1....3..2..3..3....0..3..3..0....0..3..3..2..
>..0..0..3..1....3..2..3..0....0..0..2..1....0..3..2..0....3..1..2..0..
>..0..0..3..2....0..3..1..3....0..3..3..0....0..0..3..2....3..2..0..1..
>
>Empirical for column k:
>k=1: a(n) = n + 1
>k=2: a(n) = (1/24)*n^4 + (1/12)*n^3 + (11/24)*n^2 + (53/12)*n - 6 for n>2
>k=3: [polynomial of degree 13] for n>9
>k=4: [polynomial of degree 40] for n>31
>k=5: [polynomial of degree 121] for n>98
>
> 
>rhhardin at mindspring.com
>rhhardin at att.net (either)
>
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