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A255992
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T(n,k)=Number of length n+k 0..1 arrays with at most one downstep in every k consecutive neighbor pairs
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11
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4, 8, 8, 15, 16, 16, 26, 28, 32, 32, 42, 45, 53, 64, 64, 64, 68, 80, 100, 128, 128, 93, 98, 114, 144, 188, 256, 256, 130, 136, 156, 196, 256, 354, 512, 512, 176, 183, 207, 257, 337, 451, 667, 1024, 1024, 232, 240, 268, 328, 428, 568, 796, 1256, 2048, 2048, 299, 308
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OFFSET
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1,1
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COMMENTS
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Table starts
....4....8...15...26...42...64...93..130..176..232..299..378..470..576...697
....8...16...28...45...68...98..136..183..240..308..388..481..588..710...848
...16...32...53...80..114..156..207..268..340..424..521..632..758..900..1059
...32...64..100..144..196..257..328..410..504..611..732..868.1020.1189..1376
...64..128..188..256..337..428..530..644..771..912.1068.1240.1429.1636..1862
..128..256..354..451..568..705..854.1016.1192.1383.1590.1814.2056.2317..2598
..256..512..667..796..945.1134.1352.1584.1831.2094.2374.2672.2989.3326..3684
..512.1024.1256.1413.1574.1797.2088.2419.2766.3130.3512.3913.4334.4776..5240
.1024.2048.2365.2510.2645.2848.3175.3606.4090.4592.5113.5654.6216.6800..7407
.2048.4096.4454.4448.4476.4560.4824.5294.5912.6598.7304.8031.8780.9552.10348
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LINKS
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FORMULA
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Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4)
k=4: a(n) = 2*a(n-1) -a(n-2) +3*a(n-4) -2*a(n-5)
k=5: a(n) = 2*a(n-1) -a(n-2) +4*a(n-5) -3*a(n-6)
k=6: a(n) = 2*a(n-1) -a(n-2) +5*a(n-6) -4*a(n-7)
k=7: a(n) = 2*a(n-1) -a(n-2) +6*a(n-7) -5*a(n-8)
Empirical for row n:
n=1: a(n) = (1/6)*n^3 + (1/2)*n^2 + (4/3)*n + 2
n=2: a(n) = (1/6)*n^3 + n^2 + (23/6)*n + 3
n=3: a(n) = (1/6)*n^3 + (3/2)*n^2 + (31/3)*n + 4
n=4: a(n) = (1/6)*n^3 + 2*n^2 + (143/6)*n + 6 for n>2
n=5: a(n) = (1/6)*n^3 + (5/2)*n^2 + (145/3)*n + 12 for n>3
n=6: a(n) = (1/6)*n^3 + 3*n^2 + (533/6)*n + 28 for n>4
n=7: a(n) = (1/6)*n^3 + (7/2)*n^2 + (454/3)*n + 64 for n>5
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EXAMPLE
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Some solutions for n=4 k=4
..1....1....0....0....0....0....0....1....0....0....1....0....1....0....0....0
..1....0....0....1....1....0....0....1....1....0....1....0....1....0....0....1
..1....0....1....1....1....0....1....0....0....0....1....1....0....1....0....1
..1....1....0....1....0....1....1....0....0....0....0....1....0....0....1....1
..0....1....0....0....0....1....1....1....0....1....1....1....1....0....1....1
..1....1....0....1....1....0....1....1....0....0....1....1....1....0....1....1
..1....0....0....1....1....1....1....1....1....1....1....0....1....0....1....0
..1....1....1....1....0....1....0....1....0....1....0....1....0....0....1....1
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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