A242322 T(n,k)=Number of length n+k+2 0..k arrays with every value 0..k appearing at least once in every consecutive k+3 elements, and new values 0..k introduced in order

7, 25, 13, 65, 61, 24, 140, 185, 145, 44, 266, 440, 503, 337, 81, 462, 896, 1300, 1316, 781, 149, 750, 1638, 2801, 3648, 3398, 1829, 274, 1155, 2766, 5334, 8231, 10012, 8801, 4269, 504, 1705, 4395, 9290, 16194, 23486, 27368, 23069, 9957, 927, 2431, 6655

Offset

1,1

Comments

Table starts

....7....25.....65.....140.....266.....462......750.....1155.....1705.....2431

...13....61....185.....440.....896....1638.....2766.....4395.....6655.....9691

...24...145....503....1300....2801....5334.....9290....15123....23350....34551

...44...337...1316....3648....8231...16194....28897....47931....75118...112511

...81...781...3398...10012...23486...47466....86381...145443...230647...348771

..149..1829...8801...27368...66366..137166...253674...432331...692113..1054531

..274..4269..23069...75236..187671..395166...740496..1274419..2055676..3150991

..504..9957..60197..208976..533801.1141290..2161503..3749211..6083896..9369751

..927.23233.156887..577964.1530356.3312546..6326951.11042115.18002245.27827211

.1705.54225.408962.1596216.4371836.9669270.18590776.32600811.53341987.82686971

Links

R. H. Hardin, Table of n, a(n) for n = 1..369

Formula

Empirical for column k:

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

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

k=3: [order 10]

k=4: [order 15]

k=5: [order 21]

k=6: [order 28]

Empirical for row n:

n=1: a(n) = (1/8)*n^4 + (11/12)*n^3 + (19/8)*n^2 + (31/12)*n + 1

n=2: a(n) = (5/8)*n^4 + (35/12)*n^3 + (39/8)*n^2 + (43/12)*n + 1

n=3: a(n) = (21/8)*n^4 + (89/12)*n^3 + (67/8)*n^2 + (55/12)*n + 1

n=4: a(n) = (77/8)*n^4 + (179/12)*n^3 + (103/8)*n^2 + (67/12)*n + 1

n=5: a(n) = (261/8)*n^4 + (245/12)*n^3 + (163/8)*n^2 + (79/12)*n + 1

n=6: a(n) = (845/8)*n^4 - (73/12)*n^3 + (343/8)*n^2 + (91/12)*n + 1 for n>1

n=7: a(n) = (2661/8)*n^4 - (2263/12)*n^3 + (1059/8)*n^2 + (103/12)*n + 1 for n>2

n=8: a(n) = (8237/8)*n^4 - (11701/12)*n^3 + (3879/8)*n^2 + (115/12)*n + 1 for n>3

n=9: a(n) = (25221/8)*n^4 - (46531/12)*n^3 + (14275/8)*n^2 + (127/12)*n + 1 for n>4

n=10: a(n) = (76685/8)*n^4 - (165601/12)*n^3 + (50455/8)*n^2 + (139/12)*n + 1 for n>5

n=11: a(n) = (232101/8)*n^4 - (555055/12)*n^3 + (171139/8)*n^2 + (151/12)*n + 1 for n>6

n=12: a(n) = (700397/8)*n^4 - (1794061/12)*n^3 + (561703/8)*n^2 + (163/12)*n + 1 for n>7

n=13: a(n) = (2109381/8)*n^4 - (5664667/12)*n^3 + (1798755/8)*n^2 + (175/12)*n + 1 for n>8

n=14: a(n) = (6344525/8)*n^4 - (17608249/12)*n^3 + (5657175/8)*n^2 + (187/12)*n + 1 for n>9

n=15: a(n) = (19066341/8)*n^4 - (54151687/12)*n^3 + (17559907/8)*n^2 + (199/12)*n + 1 for n>10

Example

Some solutions for n=5 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....0....1....1....1....1....1....1....1....1....0....1....0....1....1
..0....2....1....1....2....2....1....2....2....2....1....0....2....1....0....1
..2....1....2....2....3....2....0....1....3....3....2....1....3....2....2....2
..3....3....3....3....0....3....2....3....1....4....3....2....4....0....0....0
..4....0....0....0....2....2....3....4....0....2....2....3....3....3....3....3
..1....4....4....4....4....4....4....0....4....0....4....4....2....4....4....4
..2....1....2....4....1....0....1....2....3....3....0....1....0....1....4....4
..0....1....0....1....1....1....1....3....3....1....1....2....1....1....1....1
..3....2....1....2....3....4....0....1....2....1....4....0....3....3....2....2
..1....0....4....0....2....4....0....1....2....2....4....4....2....2....0....4

Crossrefs

Column 1 is A000073(n+5)

Row 1 is A001296(n+1)

Sequence in context: A286742 A322563 A065658 * A249437 A246792 A178370

Adjacent sequences: A242319 A242320 A242321 * A242323 A242324 A242325

Keyword

nonn,tabl

Author

R. H. Hardin, May 10 2014

Status

approved