OFFSET
1,1
COMMENTS
Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
.10....66....228.....580.....1230......2310......3976.......6408.......9810
.14...174....852....2780.....7170.....15834.....31304......56952......97110
.22...462...3180...13300....41730....108402....246232.....505800.....960750
.30..1206..11796...63420...242370....741090...1934856....4488696....9499590
.46..3150..43644..301780..1405530...5060706..15190840...39808584...93880710
.62..8166.160980.1433180..8139570..34523202.119174216..352838520..927352710
.94.21150.592572.6795700.47082330.235304034.934305400.3125681352.9156504150
The conjectures regarding the recursions for column k are correct (see links) - Sela Fried, Oct 29 2024.
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..9999
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024.
FORMULA
Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = 3*a(n-1) +2*a(n-2) -8*a(n-3)
k=3: a(n) = 5*a(n-1) -18*a(n-3)
k=4: a(n) = 7*a(n-1) -4*a(n-2) -32*a(n-3)
k=5: a(n) = 9*a(n-1) -10*a(n-2) -50*a(n-3)
k=6: a(n) = 11*a(n-1) -18*a(n-2) -72*a(n-3)
k=7: a(n) = 13*a(n-1) -28*a(n-2) -98*a(n-3)
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 4*n^2 + n
n=5: a(n) = n^5 + 5*n^4 + 7*n^3 + 2*n^2 - n
n=6: a(n) = n^6 + 6*n^5 + 11*n^4 + 4*n^3 - n^2 + n
n=7: a(n) = n^7 + 7*n^6 + 16*n^5 + 8*n^4 - n^3 - n
EXAMPLE
Some solutions for n=6 k=4
..2. .0. .3. .1. .1. .1. .2. .0. .1. .0. .3. .0. .3. .2. .4. .1
..4. .3. .1. .1. .2. .0. .0. .2. .0. .2. .3. .1. .4. .0. .3. .4
..3. .2. .0. .0. .0. .0. .2. .2. .1. .0. .4. .2. .4. .0. .4. .3
..3. .0. .4. .2. .1. .1. .2. .1. .3. .4. .1. .4. .2. .4. .4. .1
..1. .3. .4. .4. .0. .4. .4. .3. .4. .1. .4. .1. .2. .2. .0. .0
..0. .1. .3. .3. .4. .3. .2. .4. .1. .3. .2. .2. .1. .4. .1. .2
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Feb 27 2016
STATUS
approved