A221573 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1.

0, 0, 2, 0, 5, 2, 0, 10, 9, 4, 0, 17, 26, 25, 6, 0, 26, 59, 100, 57, 10, 0, 37, 114, 289, 342, 141, 16, 0, 50, 197, 676, 1293, 1210, 345, 26, 0, 65, 314, 1369, 3734, 5913, 4240, 853, 42, 0, 82, 471, 2500, 8991, 20944, 26911, 14898, 2097, 68, 0, 101, 674, 4225, 19014

Offset: 1

Author

R. H. Hardin Jan 20 2013

Keywords

• nonn
• tabl

Comments

Table starts

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

...2.....5......10.......17........26.........37..........50...........65

...2.....9......26.......59.......114........197.........314..........471

...4....25.....100......289.......676.......1369........2500.........4225

...6....57.....342.....1293......3734.......8991.......19014........36497

..10...141....1210.....5913.....20944......59705......145800.......317233

..16...345....4240....26911....117104.....395641.....1116400......2754635

..26...853...14898...122621....655198....2622817.....8550512.....23923281

..42..2097...52306...558547...3665306...17385993....65485386....207761745

..68..5149..183684..2544357..20505052..115249117...501533796...1804315029

.110.12633..645006.11590169.114711980..763966685..3841097940..15669633131

.178.31013.2264978.52796369.641737294.5064207645.29417832750.136083460405

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..2080

Crossrefs

Column 1 is A006355

Row 2 is A002522

Row 4 is A082044

Formula

Empirical for column k:

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

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

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

k=4: a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5) for n>6

k=5: a(n) = 5*a(n-1) +3*a(n-2) +9*a(n-4) +6*a(n-5) +3*a(n-6)

k=6: a(n) = 7*a(n-1) -4*a(n-2) +6*a(n-3) +26*a(n-4) +10*a(n-5) +16*a(n-6) +12*a(n-8)

k=7: a(n) = 7*a(n-1) +4*a(n-2) +5*a(n-3) +20*a(n-4) +20*a(n-5) +23*a(n-6) -6*a(n-7) +3*a(n-8)

Empirical for row n:

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

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

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

n=5: a(n) = 1*n^5 + 1*n^4 - 2*n^3 + 12*n^2 - 15*n + 9 for n>2

n=6: a(n) = 1*n^6 + 2*n^5 - 5*n^4 + 24*n^3 - 41*n^2 + 50*n - 31 for n>3

n=7: a(n) = 1*n^7 + 3*n^6 - 7*n^5 + 29*n^4 - 41*n^3 + 45*n^2 - 33*n + 19 for n>2