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A240783 T(n,k)=Number of nXk 0..1 arrays with no element equal to fewer vertical neighbors than horizontal neighbors, with new values 0..1 introduced in row major order.

%I A240783 #6 Jul 23 2025 11:07:24
%S A240783 1,1,2,1,3,4,1,4,11,8,1,6,20,34,16,1,9,46,97,111,32,1,14,97,305,459,
%T A240783 361,64,1,22,216,959,2167,2187,1172,128,1,35,472,3033,10150,15332,
%U A240783 10442,3809,256,1,56,1043,9581,47920,106411,108509,49861,12377,512,1,90,2296,30354
%N A240783 T(n,k)=Number of nXk 0..1 arrays with no element equal to fewer vertical neighbors than horizontal neighbors, with new values 0..1 introduced in row major order.
%C A240783 Table starts
%C A240783 ...1.....1.......1........1..........1...........1.............1..............1
%C A240783 ...2.....3.......4........6..........9..........14............22.............35
%C A240783 ...4....11......20.......46.........97.........216...........472...........1043
%C A240783 ...8....34......97......305........959........3033..........9581..........30354
%C A240783 ..16...111.....459.....2167......10150.......47920........226532........1071982
%C A240783 ..32...361....2187....15332.....106411......746346.......5228820.......36701371
%C A240783 ..64..1172...10442...108509....1120383....11677893.....121621207.....1269199948
%C A240783 .128..3809...49861...767834...11791412...182610635....2827515311....43857418181
%C A240783 .256.12377..238068..5434887..124095989..2856212777...65742420202..1515928067679
%C A240783 .512.40218.1136678.38467875.1306056075.44672652785.1528546759636.52397680462958
%H A240783 R. H. Hardin, <a href="/A240783/b240783.txt">Table of n, a(n) for n = 1..264</a>
%F A240783 Empirical for column k:
%F A240783 k=1: a(n) = 2*a(n-1)
%F A240783 k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
%F A240783 k=3: a(n) = 5*a(n-1) -a(n-2) -a(n-3) +4*a(n-4) -4*a(n-5) -3*a(n-6) +a(n-7)
%F A240783 k=4: [order 22]
%F A240783 k=5: [order 54]
%F A240783 Empirical for row n:
%F A240783 n=1: a(n) = a(n-1)
%F A240783 n=2: a(n) = 2*a(n-1) -a(n-3)
%F A240783 n=3: a(n) = 2*a(n-1) +a(n-2) -a(n-3) -a(n-5)
%F A240783 n=4: [order 15]
%F A240783 n=5: [order 30] for n>34
%F A240783 n=6: [order 94]
%e A240783 Some solutions for n=4 k=4
%e A240783 ..0..1..0..1....0..1..0..1....0..1..0..1....0..1..1..0....0..1..1..0
%e A240783 ..1..0..1..0....0..0..1..0....0..1..1..0....1..1..1..0....0..1..1..1
%e A240783 ..0..0..1..0....1..0..1..1....1..1..1..0....1..1..1..0....1..0..1..1
%e A240783 ..0..1..0..1....1..0..1..1....1..1..0..1....0..1..0..1....1..0..1..1
%Y A240783 Column 1 is A000079(n-1)
%Y A240783 Column 2 is A180762
%Y A240783 Row 2 is A001611(n+1)
%K A240783 nonn,tabl
%O A240783 1,3
%A A240783 _R. H. Hardin_, Apr 12 2014