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A232316 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every element equal to some horizontal or antidiagonal neighbor, with top left element zero.

%I A232316 #8 Jul 23 2025 07:18:13
%S A232316 2,5,5,16,24,13,52,139,115,34,169,853,1202,551,89,549,5241,14042,
%T A232316 10409,2640,233,1784,32089,164014,231454,90157,12649,610,5797,196698,
%U A232316 1905436,5142441,3815483,780922,60605,1597,18837,1205422,22161823,113293694
%N A232316 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with every element equal to some horizontal or antidiagonal neighbor, with top left element zero.
%C A232316 Table starts
%C A232316 .....2.......5.........16............52..............169.................549
%C A232316 .....5......24........139...........853.............5241...............32089
%C A232316 ....13.....115.......1202.........14042...........164014.............1905436
%C A232316 ....34.....551......10409........231454..........5142441...........113293694
%C A232316 ....89....2640......90157.......3815483........161243887..........6736602042
%C A232316 ...233...12649.....780922......62897985.......5055954492........400571676322
%C A232316 ...610...60605....6764246....1036869496.....158534446141......23818815015639
%C A232316 ..1597..290376...58591124...17092731689....4971005036586....1416315842358249
%C A232316 ..4181.1391275..507509767..281772661177..155870804492221...84217060496525106
%C A232316 .10946.6665999.4395993154.4645005493684.4887484036570530.5007720081104988709
%H A232316 R. H. Hardin, <a href="/A232316/b232316.txt">Table of n, a(n) for n = 1..390</a>
%F A232316 Empirical for column k:
%F A232316 k=1: a(n) = 3*a(n-1) -a(n-2)
%F A232316 k=2: a(n) = 5*a(n-1) -a(n-2)
%F A232316 k=3: a(n) = 10*a(n-1) -11*a(n-2) -5*a(n-3) -a(n-4)
%F A232316 k=4: a(n) = 19*a(n-1) -44*a(n-2) +43*a(n-3) -19*a(n-4) +4*a(n-5) -2*a(n-6) for n>7
%F A232316 k=5: [order 12] for n>13
%F A232316 k=6: [order 18] for n>20
%F A232316 k=7: [order 37] for n>40
%F A232316 Empirical for row n:
%F A232316 n=1: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4) for n>5
%F A232316 n=2: [order 12] for n>14
%F A232316 n=3: [order 32] for n>34
%F A232316 n=4: [order 78] for n>82
%e A232316 Some.solutions.for.n=3.k=4
%e A232316 ..0..0..0..1..1....0..0..1..1..0....0..0..0..1..1....0..0..1..1..0
%e A232316 ..0..0..1..1..1....0..1..1..0..0....0..0..0..1..1....0..1..1..0..1
%e A232316 ..1..1..0..0..0....1..0..0..1..1....0..0..1..1..1....0..0..0..1..0
%e A232316 ..1..1..1..1..1....1..1..1..0..0....0..0..0..0..0....1..1..1..0..0
%Y A232316 Column 1 is A001519(n+1)
%Y A232316 Column 2 is A004254(n+1)
%K A232316 nonn,tabl
%O A232316 1,1
%A A232316 _R. H. Hardin_, Nov 22 2013