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%I #6 Aug 19 2022 13:37:36
%S 4,8,8,16,16,16,32,32,32,32,63,64,64,64,64,120,124,128,128,128,128,
%T 219,229,245,256,256,256,256,382,402,442,484,512,512,512,512,638,673,
%U 753,856,956,1024,1024,1024,1024,1024,1080,1220,1424,1656,1888,2048,2048,2048
%N T(n,k) = Number of length n+k 0..1 arrays with at most two downsteps in every k consecutive neighbor pairs.
%C Table starts
%C ....4....8...16....32....63...120...219...382....638...1024...1586...2380
%C ....8...16...32....64...124...229...402...673...1080...1670...2500...3638
%C ...16...32...64...128...245...442...753..1220...1894...2836...4118...5824
%C ...32...64..128...256...484...856..1424..2249...3402...4965...7032...9710
%C ...64..128..256...512...956..1656..2693..4158...6153...8792..12202..16524
%C ..128..256..512..1024..1888..3204..5088..7677..11120..15579..21230..28264
%C ..256..512.1024..2048..3728..6192..9613.14168..20075..27566..36888..48304
%C ..512.1024.2048..4096..7362.11955.18104.26117..36218..48738..64024..82440
%C .1024.2048.4096..8192.14539.23088.34013.47858..65130..86008.110976.140536
%C .2048.4096.8192.16384.28712.44617.63928.87338.116104.150906.191620.238932
%H R. H. Hardin, <a href="/A256816/b256816.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)
%F k=4: a(n) = 2*a(n-1)
%F k=5: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4) +2*a(n-5) -a(n-6)
%F k=6: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4) +3*a(n-6) -2*a(n-7) -6*a(n-9) +4*a(n-10)
%F k=7: [order 15]
%F Empirical for row n:
%F n=1: a(n) = (1/120)*n^5 + (1/8)*n^3 + (1/2)*n^2 + (41/30)*n + 2
%F n=2: a(n) = (1/120)*n^5 + (1/24)*n^4 + (3/8)*n^3 - (1/24)*n^2 + (277/60)*n + 3
%F n=3: a(n) = (1/120)*n^5 + (1/12)*n^4 + (31/24)*n^3 - (31/12)*n^2 + (66/5)*n + 4
%F n=4: [polynomial of degree 5] for n>2
%F n=5: [polynomial of degree 5] for n>3
%F n=6: [polynomial of degree 5] for n>4
%F n=7: [polynomial of degree 5] for n>5
%e Some solutions for n=4, k=4
%e ..1....1....0....0....0....0....1....0....0....0....0....0....1....0....0....1
%e ..0....0....1....1....0....1....0....1....1....0....0....0....1....0....0....1
%e ..1....1....0....1....0....0....1....0....1....1....1....1....0....1....0....1
%e ..0....1....1....1....0....1....1....1....1....1....0....0....1....1....0....0
%e ..0....1....0....0....1....1....1....1....0....0....1....1....1....1....0....0
%e ..0....1....1....0....1....1....0....0....0....1....0....0....0....1....0....1
%e ..0....0....1....0....1....1....1....0....0....1....1....1....0....0....0....0
%e ..0....1....0....1....1....1....0....1....0....1....0....1....0....1....0....1
%Y Column 1 is A000079(n+1).
%Y Column 2 is A000079(n+2).
%Y Column 3 is A000079(n+3).
%Y Column 4 is A000079(n+4).
%Y Row 1 is A006261(n+1).
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Apr 10 2015
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