login
T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by more than one.
11

%I #4 Mar 01 2016 07:58:02

%S 2,3,4,4,9,8,5,16,27,16,6,25,64,79,32,7,36,125,250,229,64,8,49,216,

%T 613,964,659,128,9,64,343,1276,2969,3680,1889,256,10,81,512,2371,7456,

%U 14239,13946,5401,512,11,100,729,4054,16237,43184,67763,52562,15419,1024,12

%N T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by more than one.

%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 ...16....79....250.....613.....1276......2371......4054.......6505.......9928

%C ...32...229....964....2969.....7456.....16237.....31844......57649......97984

%C ...64...659...3680...14239....43184....110339....248464.....507935.....962144

%C ..128..1889..13946...67763...248324....745013...1927694....4453031....9406088

%C ..256..5401..52562..320495..1419502...5003189..14883506...38870827...91601150

%C ..512.15419.197288.1508267..8074172..33444515.114432704..338024963..889023812

%C .1024.43977.738190.7069055.45734140.222678103.876609410.2929722175.8602245520

%H R. H. Hardin, <a href="/A269583/b269583.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)

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

%F k=3: a(n) = 9*a(n-1) -24*a(n-2) +9*a(n-3) +26*a(n-4) +3*a(n-5)

%F k=4: a(n) = 16*a(n-1) -93*a(n-2) +220*a(n-3) -112*a(n-4) -192*a(n-5) -2*a(n-6) +8*a(n-7)

%F k=5: [order 7]

%F k=6: [order 9]

%F k=7: [order 9]

%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 + 5*n^2 + 5*n + 1

%F n=5: a(n) = n^5 + 5*n^4 + 7*n^3 + 12*n^2 + 6*n + 1

%F n=6: a(n) = n^6 + 6*n^5 + 9*n^4 + 22*n^3 + 15*n^2 + 12*n - 1

%F n=7: a(n) = n^7 + 7*n^6 + 11*n^5 + 35*n^4 + 28*n^3 + 42*n^2 + 5*n - 1

%e Some solutions for n=6 k=4

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

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

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

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

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

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

%Y Column 1 is A000079.

%Y Column 3 is A269489.

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

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

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

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Mar 01 2016