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A269467
T(n,k)=Number of length-n 0..k arrays with no repeated value equal to the previous repeated value.
12
2, 3, 4, 4, 9, 6, 5, 16, 24, 10, 6, 25, 60, 66, 14, 7, 36, 120, 228, 174, 22, 8, 49, 210, 580, 852, 462, 30, 9, 64, 336, 1230, 2780, 3180, 1206, 46, 10, 81, 504, 2310, 7170, 13300, 11796, 3150, 62, 11, 100, 720, 3976, 15834, 41730, 63420, 43644, 8166, 94, 12, 121
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
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
Column 1 is A027383.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).
Sequence in context: A268944 A269537 A269678 * A228461 A267471 A268457
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Feb 27 2016
STATUS
approved