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

1,3

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

Links

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

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

Example

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

Crossrefs

Column 1 is A006355

Row 2 is A002522

Row 4 is A082044

Sequence in context: A192426 A075603 A264357 * A332453 A359359 A066283

Adjacent sequences: A221570 A221571 A221572 * A221574 A221575 A221576

Keyword

nonn,tabl

Author

R. H. Hardin Jan 20 2013

Status

approved