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%I #4 May 08 2014 17:36:23
%S 3,6,5,10,12,8,15,22,22,13,21,35,43,40,21,28,51,71,82,74,34,36,70,106,
%T 139,157,136,55,45,92,148,211,271,304,250,89,55,117,197,298,416,531,
%U 586,460,144,66,145,253,400,592,821,1047,1129,846,233,78,176,316,517,799
%N T(n,k)=Number of length n+k+1 0..k arrays with every value 0..k appearing at least once in every consecutive k+2 elements, and new values 0..k introduced in order
%C Table starts
%C ...3....6...10...15....21....28....36....45....55....66....78....91...105
%C ...5...12...22...35....51....70....92...117...145...176...210...247...287
%C ...8...22...43...71...106...148...197...253...316...386...463...547...638
%C ..13...40...82..139...211...298...400...517...649...796...958..1135..1327
%C ..21...74..157..271...416...592...799..1037..1306..1606..1937..2299..2692
%C ..34..136..304..531...821..1174..1590..2069..2611..3216..3884..4615..5409
%C ..55..250..586.1047..1626..2332..3165..4125..5212..6426..7767..9235.10830
%C ..89..460.1129.2059..3231..4642..6308..8229.10405.12836.15522.18463.21659
%C .144..846.2176.4047..6411..9256.12587.16429.20782.25646.31021.36907.43304
%C .233.1556.4195.7955.12716.18442.25138.32821.41527.51256.62008.73783.86581
%H R. H. Hardin, <a href="/A242239/b242239.txt">Table of n, a(n) for n = 1..731</a>
%F Empirical for column k:
%F k=1: a(n) = a(n-1) +a(n-2)
%F k=2: a(n) = a(n-1) +a(n-2) +a(n-3)
%F k=3: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4)
%F k=4: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5)
%F k=5: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6)
%F k=6: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7)
%F k=7: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8)
%F k=8: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9)
%F k=9: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9) +a(n-10)
%F Empirical for row n:
%F n=1: a(n) = (1/2)*n^2 + (3/2)*n + 1
%F n=2: a(n) = (3/2)*n^2 + (5/2)*n + 1
%F n=3: a(n) = (7/2)*n^2 + (7/2)*n + 1
%F n=4: a(n) = (15/2)*n^2 + (9/2)*n + 1
%F n=5: a(n) = (31/2)*n^2 + (11/2)*n + 1 for n>1
%F n=6: a(n) = (63/2)*n^2 + (13/2)*n + 1 for n>2
%F n=7: a(n) = (127/2)*n^2 + (15/2)*n + 1 for n>3
%F n=8: a(n) = (255/2)*n^2 + (17/2)*n + 1 for n>4
%F n=9: a(n) = (511/2)*n^2 + (19/2)*n + 1 for n>5
%F n=10: a(n) = (1023/2)*n^2 + (21/2)*n + 1 for n>6
%F n=11: a(n) = (2047/2)*n^2 + (23/2)*n + 1 for n>7
%F n=12: a(n) = (4095/2)*n^2 + (25/2)*n + 1 for n>8
%F n=13: a(n) = (8191/2)*n^2 + (27/2)*n + 1 for n>9
%F n=14: a(n) = (16383/2)*n^2 + (29/2)*n + 1 for n>10
%F n=15: a(n) = (32767/2)*n^2 + (31/2)*n + 1 for n>11
%F Empirical large-k generalization, for k>n-4: T(n,k) = ((2^n-1)/2)*k^2 + ((2*n+1)/2)*k + 1
%F Empirical recurrence generalization, for column k: a(n) = sum {i in 1..k+1} a(n-i)
%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....1....1....1....1....1....1....1....1....1....1....1....0....1....1
%e ..2....2....0....2....2....1....0....0....2....0....2....2....0....1....2....0
%e ..1....3....2....3....3....2....2....2....3....2....3....3....2....2....3....2
%e ..3....4....3....4....4....3....3....3....0....3....0....0....3....3....4....3
%e ..4....0....4....1....0....4....4....4....4....4....4....4....4....4....1....4
%e ..0....2....2....0....1....0....2....0....2....1....2....1....0....2....0....1
%e ..2....1....1....0....2....1....1....1....1....0....1....2....1....0....2....0
%e ..2....3....0....2....3....0....0....1....0....0....3....4....2....1....3....4
%e ..1....0....0....3....0....2....2....2....3....2....2....3....3....1....0....2
%Y Column 1 is A000045(n+3)
%Y Column 2 is A196700(n+3)
%Y Row 1 is A000217(n+1)
%Y Row 2 is A000326(n+1)
%Y Row 3 is A069099(n+1)
%Y Row 4 is A220083
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, May 08 2014
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