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A208637 T(n,k)=Number of nXk 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

%I A208637 #7 Jul 22 2025 21:27:15
%S A208637 1,2,2,4,5,4,8,11,13,8,16,23,32,34,16,32,47,71,95,89,32,64,95,150,225,
%T A208637 284,233,64,128,191,309,494,722,851,610,128,256,383,628,1042,1652,
%U A208637 2331,2552,1597,256,512,767,1267,2149,3577,5572,7548,7655,4181,512,1024,1535
%N A208637 T(n,k)=Number of nXk 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.
%C A208637 Table starts
%C A208637 ...1....2....4.....8....16.....32.....64....128.....256.....512....1024
%C A208637 ...2....5...11....23....47.....95....191....383.....767....1535....3071
%C A208637 ...4...13...32....71...150....309....628...1267....2546....5105...10224
%C A208637 ...8...34...95...225...494...1042...2149...4375....8840...17784...35687
%C A208637 ..16...89..284...722..1652...3577...7504..15448...31440...63543..127884
%C A208637 ..32..233..851..2331..5572..12404..26508..55260..113427..230559..465773
%C A208637 ..64..610.2552..7548.18888..43284..94320.199299..412962..844943.1714680
%C A208637 .128.1597.7655.24476.64216.151656.337227.722733.1512764.3117620.6359210
%H A208637 R. H. Hardin, <a href="/A208637/b208637.txt">Table of n, a(n) for n = 1..1193</a>
%F A208637 Empirical for column k:
%F A208637 k=1: a(n) = 2*a(n-1)
%F A208637 k=2: a(n) = 3*a(n-1) -a(n-2)
%F A208637 k=3: a(n) = 4*a(n-1) -3*a(n-2)
%F A208637 k=4: a(n) = 5*a(n-1) -6*a(n-2) +a(n-3)
%F A208637 k=5: a(n) = 6*a(n-1) -10*a(n-2) +4*a(n-3)
%F A208637 k=6: a(n) = 7*a(n-1) -15*a(n-2) +10*a(n-3) -a(n-4)
%F A208637 k=7: a(n) = 8*a(n-1) -21*a(n-2) +20*a(n-3) -5*a(n-4)
%F A208637 Empirical for row n:
%F A208637 n=1: a(k)=2*a(k-1)
%F A208637 n=2: a(k)=3*a(k-1)-2*a(k-2)
%F A208637 n=3: a(k)=4*a(k-1)-5*a(k-2)+2*a(k-3)
%F A208637 n=4: a(k)=5*a(k-1)-9*a(k-2)+7*a(k-3)-2*a(k-4) for k>5
%F A208637 n=5: a(k)=6*a(k-1)-14*a(k-2)+16*a(k-3)-9*a(k-4)+2*a(k-5) for k>7
%F A208637 n=6: a(k)=7*a(k-1)-20*a(k-2)+30*a(k-3)-25*a(k-4)+11*a(k-5)-2*a(k-6) for k>9
%F A208637 n=7: a(k)=8*a(k-1)-27*a(k-2)+50*a(k-3)-55*a(k-4)+36*a(k-5)-13*a(k-6)+2*a(k-7) for k>11
%e A208637 Some solutions for n=4 k=3
%e A208637 ..0..0..0....0..0..1....0..1..1....0..0..1....0..0..1....0..1..0....0..1..0
%e A208637 ..1..1..0....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1
%e A208637 ..0..1..1....0..1..0....0..1..0....0..1..0....1..0..0....0..1..0....1..0..1
%e A208637 ..0..0..1....1..0..1....0..1..0....0..1..0....1..1..0....1..0..1....1..0..0
%Y A208637 Column 2 is A001519(n+1)
%Y A208637 Column 3 is A199109(n-1)
%Y A208637 Row 2 is A052940(n-1)
%K A208637 nonn,tabl
%O A208637 1,2
%A A208637 _R. H. Hardin_ Feb 29 2012