A221524 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 2 or more

0, 0, 0, 0, 2, 0, 0, 6, 2, 0, 0, 12, 10, 4, 0, 0, 20, 30, 36, 6, 0, 0, 30, 68, 144, 94, 10, 0, 0, 42, 130, 400, 536, 274, 16, 0, 0, 56, 222, 900, 1940, 2172, 768, 26, 0, 0, 72, 350, 1764, 5368, 9982, 8544, 2182, 42, 0, 0, 90, 520, 3136, 12458, 33380, 50400, 33960, 6170, 68, 0, 0

Offset

1,5

Comments

Table starts

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

.0...2......6......12........20.........30..........42..........56...........72

.0...2.....10......30........68........130.........222.........350..........520

.0...4.....36.....144.......400........900........1764........3136.........5184

.0...6.....94.....536......1940.......5368.......12458.......25544........47776

.0..10....274....2172......9982......33380.......90684......212812.......447962

.0..16....768....8544.....50400.....205080......654864.....1763328......4184064

.0..26...2182...33960....256018....1264378.....4738970....14629962.....39113752

.0..42...6170..134480...1297924....7787228....34274630...121342546....365574840

.0..68..17476..533248...6584320...47975704...247928860..1006508448...3416978176

.0.110..49470.2113456..33394958..295543282..1793345580..8348594292..31937713030

.0.178.140066.8377808.169387004.1820672982.12971955294.69248649436.298515152986

Links

R. H. Hardin, Table of n, a(n) for n = 1..1518

Formula

Empirical for column k:

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

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

k=4: a(n) = 2*a(n-1) +6*a(n-2) +6*a(n-3) +4*a(n-4) +4*a(n-6)

k=5: a(n) = 2*a(n-1) +11*a(n-2) +20*a(n-3) +17*a(n-4) -3*a(n-5) +a(n-6)

k=6: a(n) = 3*a(n-1) +14*a(n-2) +29*a(n-3) +28*a(n-4) +a(n-5) +27*a(n-6) +8*a(n-7) +2*a(n-8)

k=7: a(n) = 3*a(n-1) +21*a(n-2) +58*a(n-3) +79*a(n-4) +32*a(n-5) +23*a(n-6) +4*a(n-7) +8*a(n-8)

Empirical for row n:

n=2: a(n) = n^2 - n

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

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

n=5: a(n) = n^5 - n^4 - 10*n^3 + 38*n^2 - 60*n + 40 for n>2

n=6: a(n) = n^6 - 20*n^4 + 83*n^3 - 182*n^2 + 236*n - 148 for n>3

n=7: a(n) = n^7 + n^6 - 29*n^5 + 109*n^4 - 204*n^3 + 202*n^2 - 80*n for n>2

Example

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

Crossrefs

Column 2 is A006355

Row 2 is A002378(n-1)

Row 3 is A034262(n-1)

Row 4 is A035287

Sequence in context: A339942 A345366 A278720 * A193009 A045866 A257549

Adjacent sequences: A221521 A221522 A221523 * A221525 A221526 A221527

Keyword

nonn,tabl

Author

R. H. Hardin Jan 19 2013

Status

approved