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T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by plus or minus one modulo k+1.
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%I #4 Mar 03 2016 13:36:33

%S 2,3,4,4,9,8,5,16,27,14,6,25,64,75,24,7,36,125,248,201,40,8,49,216,

%T 615,944,525,66,9,64,343,1284,2995,3544,1347,108,10,81,512,2387,7584,

%U 14465,13168,3411,176,11,100,729,4080,16541,44556,69405,48536,8553,286,12,121

%N T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by plus or minus one modulo k+1.

%C Table starts

%C ...2.....3......4.......5........6.........7.........8..........9.........10

%C ...4.....9.....16......25.......36........49........64.........81........100

%C ...8....27.....64.....125......216.......343.......512........729.......1000

%C ..14....75....248.....615.....1284......2387......4080.......6543.......9980

%C ..24...201....944....2995.....7584.....16541.....32416......58599......99440

%C ..40...525...3544...14465....44556....114205....256880.....523809.....989380

%C ..66..1347..13168...69405...260616....786079...2031072....4674393....9831160

%C .108..3411..48536..331255..1518804...5396363..16027696...41651631...97576460

%C .176..8553.177776.1574195..8823984..36961757.126262688..370652391..967466240

%C .286.21285.647896.7454385.51132636.252671461.993181680.3294529281.9583467220

%H R. H. Hardin, <a href="/A269690/b269690.txt">Table of n, a(n) for n = 1..9999</a>

%F Empirical for column k:

%F k=1: a(n) = 2*a(n-1) -a(n-3)

%F k=2: a(n) = 4*a(n-1) -3*a(n-2) -2*a(n-3)

%F k=3: a(n) = 6*a(n-1) -7*a(n-2) -6*a(n-3)

%F k=4: a(n) = 8*a(n-1) -13*a(n-2) -12*a(n-3)

%F k=5: a(n) = 10*a(n-1) -21*a(n-2) -20*a(n-3)

%F k=6: a(n) = 12*a(n-1) -31*a(n-2) -30*a(n-3)

%F k=7: a(n) = 14*a(n-1) -43*a(n-2) -42*a(n-3)

%F Empirical for row n:

%F n=1: a(n) = n + 1

%F n=2: a(n) = n^2 + 2*n + 1

%F n=3: a(n) = n^3 + 3*n^2 + 3*n + 1

%F n=4: a(n) = n^4 + 4*n^3 + 6*n^2 + 2*n - 1 for n>1

%F n=5: a(n) = n^5 + 5*n^4 + 10*n^3 + 4*n^2 - 3*n - 1 for n>1

%F n=6: a(n) = n^6 + 6*n^5 + 15*n^4 + 8*n^3 - 7*n^2 - 4*n + 1 for n>1

%F n=7: a(n) = n^7 + 7*n^6 + 21*n^5 + 15*n^4 - 13*n^3 - 11*n^2 + 3*n + 1 for n>1

%e Some solutions for n=6 k=4

%e ..0. .3. .4. .2. .4. .3. .3. .0. .3. .0. .4. .1. .1. .3. .0. .0

%e ..1. .2. .1. .2. .0. .0. .0. .4. .0. .1. .1. .3. .3. .2. .2. .2

%e ..3. .3. .0. .4. .2. .2. .2. .4. .0. .0. .2. .3. .1. .1. .1. .2

%e ..2. .1. .0. .3. .2. .0. .2. .3. .0. .3. .0. .4. .3. .0. .3. .4

%e ..4. .0. .3. .4. .4. .1. .4. .4. .3. .2. .3. .0. .3. .2. .3. .1

%e ..4. .2. .1. .3. .0. .4. .2. .3. .2. .2. .1. .1. .1. .1. .3. .0

%Y Column 1 is A019274(n+2).

%Y Row 1 is A000027(n+1).

%Y Row 2 is A000290(n+1).

%Y Row 3 is A000578(n+1).

%Y Row 4 is A246767(n+2) for n>1.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Mar 03 2016