A221463 T(n,k)=Number of 0..k arrays of length n with each element unequal to at least one neighbor, starting with 0

0, 0, 1, 0, 2, 1, 0, 3, 4, 2, 0, 4, 9, 12, 3, 0, 5, 16, 36, 32, 5, 0, 6, 25, 80, 135, 88, 8, 0, 7, 36, 150, 384, 513, 240, 13, 0, 8, 49, 252, 875, 1856, 1944, 656, 21, 0, 9, 64, 392, 1728, 5125, 8960, 7371, 1792, 34, 0, 10, 81, 576, 3087, 11880, 30000, 43264, 27945, 4896, 55, 0, 11

Offset

1,5

Comments

Table starts

..0.....0.......0........0.........0..........0..........0...........0

..1.....2.......3........4.........5..........6..........7...........8

..1.....4.......9.......16........25.........36.........49..........64

..2....12......36.......80.......150........252........392.........576

..3....32.....135......384.......875.......1728.......3087........5120

..5....88.....513.....1856......5125......11880......24353.......45568

..8...240....1944.....8960.....30000......81648.....192080......405504

.13...656....7371....43264....175625.....561168....1515031.....3608576

.21..1792...27945...208896...1028125....3856896...11949777....32112640

.34..4896..105948..1008640...6018750...26508384...94253656...285769728

.55.13376..401679..4870144..35234375..182191680..743424031..2543058944

.89.36544.1522881.23515136.206265625.1252200384.5863743809.22630629376

Links

R. H. Hardin, Table of n, a(n) for n = 1..10000

Formula

Recursion for column k:

k=1: a(n) = a(n-1) +a(n-2)

k=2: a(n) = 2*a(n-1) +2*a(n-2)

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

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

k=5: a(n) = 5*a(n-1) +5*a(n-2)

k=6: a(n) = 6*a(n-1) +6*a(n-2)

k=7: a(n) = 7*a(n-1) +7*a(n-2)

Empirical for row n:

n=2: a(k) = 1*k

n=3: a(k) = 1*k^2

n=4: a(k) = 1*k^3 + 1*k^2

n=5: a(k) = 1*k^4 + 2*k^3

n=6: a(k) = 1*k^5 + 3*k^4 + 1*k^3

n=7: a(k) = 1*k^6 + 4*k^5 + 3*k^4

n=8: a(k) = 1*k^7 + 5*k^6 + 6*k^5 + 1*k^4

n=9: a(k) = 1*k^8 + 6*k^7 + 10*k^6 + 4*k^5

n=10: a(k) = 1*k^9 + 7*k^8 + 15*k^7 + 10*k^6 + 1*k^5

n=11: a(k) = 1*k^10 + 8*k^9 + 21*k^8 + 20*k^7 + 5*k^6

n=12: a(k) = 1*k^11 + 9*k^10 + 28*k^9 + 35*k^8 + 15*k^7 + 1*k^6

n=13: a(k) = 1*k^12 + 10*k^11 + 36*k^10 + 56*k^9 + 35*k^8 + 6*k^7

n=14: a(k) = 1*k^13 + 11*k^12 + 45*k^11 + 84*k^10 + 70*k^9 + 21*k^8 + 1*k^7

n=15: a(k) = 1*k^14 + 12*k^13 + 55*k^12 + 120*k^11 + 126*k^10 + 56*k^9 + 7*k^8

Apparently then T(n,k) = sum { binomial(n-2-i,i)*k^(n-1-i) , 0<=2*i<=n-2 }.

The formula reduces to T(n,k) = [4*k^(n-1)*(1+G)^(2*n-2) +4^n] /[2^(n+1) *G *(1+G)^(n-1)] for even n and to T(n,k) = [4*k^(n-1) *(1+G)^(2*n-2) -4^n] /[2^(n+1) *G *(1+G)^(n-1)] for odd n, where G=sqrt(1+4/k). - R. J. Mathar, Jan 21 2013

Example

Some solutions for n=6 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....2....1....3....3....2....2....1....4....2....3....1....3....2....3....3
..1....4....0....1....2....4....3....0....2....0....3....3....2....0....0....4
..0....0....3....4....2....3....2....1....0....3....2....4....3....3....4....1
..1....2....1....2....1....0....3....4....0....1....3....1....3....0....2....0
..0....3....2....3....3....1....4....0....4....2....0....0....0....4....0....3

Crossrefs

Column 1 is A000045(n-1)

Column 2 is A028860(n+1)

Column 3 is A106435(n-1)

Column 4 is A094013

Column 5 is A106565(n-1)

Row 2 is A000027

Row 3 is A000290

Row 4 is A011379

Sequence in context: A106378 A094301 A221542 * A135488 A099493 A088523

Adjacent sequences: A221460 A221461 A221462 * A221464 A221465 A221466

Keyword

nonn,tabl

Author

R. H. Hardin, general recursion proved by Robert Israel in the Sequence Fans Mailing List, Jan 17 2013

Status

approved