[seqfan] Long Transients before Polynomial

Ron Hardin rhhardin at att.net
Tue Mar 25 20:46:51 CET 2014


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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