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T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, and adjacent elements not equal modulo three (with -1 modulo 3 = 2)
12

%I #5 Mar 31 2012 12:36:37

%S 1,1,2,1,4,6,1,4,12,8,1,6,24,24,14,1,8,42,72,82,32,1,8,60,152,256,232,

%T 56,1,10,84,256,804,1312,654,100,1,12,114,448,1836,5016,5206,2044,204,

%U 1,12,144,680,3196,12872,24864,21208,6096,388,1,14,180,952,6064,29864,77874

%N T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, and adjacent elements not equal modulo three (with -1 modulo 3 = 2)

%C Table starts

%C ...1.....1......1.......1........1.........1.........1..........1..........1

%C ...2.....4......4.......6........8.........8........10.........12.........12

%C ...6....12.....24......42.......60........84.......114........144........180

%C ...8....24.....72.....152......256.......448.......680........952.......1384

%C ..14....82....256.....804.....1836......3196......6064......10276......14846

%C ..32...232...1312....5016....12872.....29864.....62776.....114768.....200520

%C ..56...654...5206...24864....77874....216530....518560....1071202....2114394

%C .100..2044..21208..139148...547604...1699268...4854740...11588992...24551100

%C .204..6096..97668..814776..3784512..14546928..47329800..125461824..306360336

%C .388.18564.422052.4509164.25525476.116482068.436295060.1308549932.3582143596

%H R. H. Hardin, <a href="/A199909/b199909.txt">Table of n, a(n) for n = 1..871</a>

%F Empirical for rows:

%F T(1,k)=1

%F T(2,k)=a(k-1)+a(k-3)-a(k-4)

%F T(3,k)=2*a(k-1)-a(k-2)+a(k-3)-2*a(k-4)+a(k-5)

%F T(4,k)=a(k-1)+3*a(k-3)-3*a(k-4)-3*a(k-6)+3*a(k-7)+a(k-9)-a(k-10)

%F T(5,k)=a(k-1)+4*a(k-3)-4*a(k-4)-6*a(k-6)+6*a(k-7)+4*a(k-9)-4*a(k-10)-a(k-12)+a(k-13)

%F T(6,k)=2*a(k-1)-a(k-2)+4*a(k-3)-8*a(k-4)+4*a(k-5)-6*a(k-6)+12*a(k-7)-6*a(k-8)+4*a(k-9)-8*a(k-10)+4*a(k-11)-a(k-12)+2*a(k-13)-a(k-14)

%F T(7,k)=a(k-1)+6*a(k-3)-6*a(k-4)-15*a(k-6)+15*a(k-7)+20*a(k-9)-20*a(k-10)-15*a(k-12)+15*a(k-13)+6*a(k-15)-6*a(k-16)-a(k-18)+a(k-19)

%e Some solutions for n=7 k=6

%e .-6...-3....4...-6...-3....4....4...-6....4....3....0....3...-6...-6....0....4

%e .-4....2....2...-4...-4....3...-1...-1....5....2....4....4....4....5...-1...-6

%e ..4...-5....0...-3...-3....1....0....3...-5....4....0...-3...-6...-3...-5....4

%e .-4....6...-1....5....2...-6...-2....1...-4....0...-2...-1....1....1....0...-1

%e ..6....5....0....4....3....5...-6...-1...-6...-4...-4...-5...-1...-4...-2....0

%e .-2...-6....1....6....5...-3....2....6....2...-3....6....5....6....1....6...-4

%e ..6....1...-6...-2....0...-4....3...-2....4...-2...-4...-3....2....6....2....3

%Y Column 1 is A199697

%Y Row 2 is A063200(n+2)

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_ Nov 11 2011