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%I #4 Mar 13 2015 12:41:38
%S 4,8,8,15,16,16,26,28,32,32,42,45,53,64,64,64,68,80,100,128,128,93,98,
%T 114,144,188,256,256,130,136,156,196,256,354,512,512,176,183,207,257,
%U 337,451,667,1024,1024,232,240,268,328,428,568,796,1256,2048,2048,299,308
%N T(n,k)=Number of length n+k 0..1 arrays with at most one downstep in every k consecutive neighbor pairs
%C Table starts
%C ....4....8...15...26...42...64...93..130..176..232..299..378..470..576...697
%C ....8...16...28...45...68...98..136..183..240..308..388..481..588..710...848
%C ...16...32...53...80..114..156..207..268..340..424..521..632..758..900..1059
%C ...32...64..100..144..196..257..328..410..504..611..732..868.1020.1189..1376
%C ...64..128..188..256..337..428..530..644..771..912.1068.1240.1429.1636..1862
%C ..128..256..354..451..568..705..854.1016.1192.1383.1590.1814.2056.2317..2598
%C ..256..512..667..796..945.1134.1352.1584.1831.2094.2374.2672.2989.3326..3684
%C ..512.1024.1256.1413.1574.1797.2088.2419.2766.3130.3512.3913.4334.4776..5240
%C .1024.2048.2365.2510.2645.2848.3175.3606.4090.4592.5113.5654.6216.6800..7407
%C .2048.4096.4454.4448.4476.4560.4824.5294.5912.6598.7304.8031.8780.9552.10348
%H R. H. Hardin, <a href="/A255992/b255992.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical for column k:
%F k=1: a(n) = 2*a(n-1)
%F k=2: a(n) = 2*a(n-1)
%F k=3: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4)
%F k=4: a(n) = 2*a(n-1) -a(n-2) +3*a(n-4) -2*a(n-5)
%F k=5: a(n) = 2*a(n-1) -a(n-2) +4*a(n-5) -3*a(n-6)
%F k=6: a(n) = 2*a(n-1) -a(n-2) +5*a(n-6) -4*a(n-7)
%F k=7: a(n) = 2*a(n-1) -a(n-2) +6*a(n-7) -5*a(n-8)
%F Empirical for row n:
%F n=1: a(n) = (1/6)*n^3 + (1/2)*n^2 + (4/3)*n + 2
%F n=2: a(n) = (1/6)*n^3 + n^2 + (23/6)*n + 3
%F n=3: a(n) = (1/6)*n^3 + (3/2)*n^2 + (31/3)*n + 4
%F n=4: a(n) = (1/6)*n^3 + 2*n^2 + (143/6)*n + 6 for n>2
%F n=5: a(n) = (1/6)*n^3 + (5/2)*n^2 + (145/3)*n + 12 for n>3
%F n=6: a(n) = (1/6)*n^3 + 3*n^2 + (533/6)*n + 28 for n>4
%F n=7: a(n) = (1/6)*n^3 + (7/2)*n^2 + (454/3)*n + 64 for n>5
%e Some solutions for n=4 k=4
%e ..1....1....0....0....0....0....0....1....0....0....1....0....1....0....0....0
%e ..1....0....0....1....1....0....0....1....1....0....1....0....1....0....0....1
%e ..1....0....1....1....1....0....1....0....0....0....1....1....0....1....0....1
%e ..1....1....0....1....0....1....1....0....0....0....0....1....0....0....1....1
%e ..0....1....0....0....0....1....1....1....0....1....1....1....1....0....1....1
%e ..1....1....0....1....1....0....1....1....0....0....1....1....1....0....1....1
%e ..1....0....0....1....1....1....1....1....1....1....1....0....1....0....1....0
%e ..1....1....1....1....0....1....0....1....0....1....0....1....0....0....1....1
%Y Column 1 is A000079(n+1)
%Y Column 2 is A000079(n+2)
%Y Column 3 is A118870(n+3)
%Y Row 1 is A000125(n+1)
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Mar 13 2015
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