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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
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%I #4 May 10 2014 18:49:56

%S 7,25,13,65,61,24,140,185,145,44,266,440,503,337,81,462,896,1300,1316,

%T 781,149,750,1638,2801,3648,3398,1829,274,1155,2766,5334,8231,10012,

%U 8801,4269,504,1705,4395,9290,16194,23486,27368,23069,9957,927,2431,6655

%N 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

%C Table starts

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

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

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

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

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

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

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

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

%C ..927.23233.156887..577964.1530356.3312546..6326951.11042115.18002245.27827211

%C .1705.54225.408962.1596216.4371836.9669270.18590776.32600811.53341987.82686971

%H R. H. Hardin, <a href="/A242322/b242322.txt">Table of n, a(n) for n = 1..369</a>

%F Empirical for column k:

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

%F 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)

%F k=3: [order 10]

%F k=4: [order 15]

%F k=5: [order 21]

%F k=6: [order 28]

%F Empirical for row n:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%e Some solutions for n=5 k=4

%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

%e ..1....1....0....1....1....1....1....1....1....1....1....0....1....0....1....1

%e ..0....2....1....1....2....2....1....2....2....2....1....0....2....1....0....1

%e ..2....1....2....2....3....2....0....1....3....3....2....1....3....2....2....2

%e ..3....3....3....3....0....3....2....3....1....4....3....2....4....0....0....0

%e ..4....0....0....0....2....2....3....4....0....2....2....3....3....3....3....3

%e ..1....4....4....4....4....4....4....0....4....0....4....4....2....4....4....4

%e ..2....1....2....4....1....0....1....2....3....3....0....1....0....1....4....4

%e ..0....1....0....1....1....1....1....3....3....1....1....2....1....1....1....1

%e ..3....2....1....2....3....4....0....1....2....1....4....0....3....3....2....2

%e ..1....0....4....0....2....4....0....1....2....2....4....4....2....2....0....4

%Y Column 1 is A000073(n+5)

%Y Row 1 is A001296(n+1)

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, May 10 2014