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

Author

R. H. Hardin Jan 19 2013

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Column 2 is A006355

Row 2 is A002378(n-1)

Row 3 is A034262(n-1)

Row 4 is A035287

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