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

Author

R. H. Hardin, May 10 2014

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

Crossrefs

Column 1 is A000073(n+5)

Row 1 is A001296(n+1)

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