A269467 T(n,k)=Number of length-n 0..k arrays with no repeated value equal to the previous repeated value.

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

Author

R. H. Hardin, Feb 27 2016

Keywords

• nonn
• tabl

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.

Examples

			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
		

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.

Crossrefs

Column 1 is A027383.

Row 1 is A000027(n+1).

Row 2 is A000290(n+1).

Row 3 is A007531(n+2).

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