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%I #4 Jan 22 2013 06:31:05
%S 0,0,2,0,2,4,0,2,5,8,0,2,5,14,16,0,2,5,20,40,32,0,2,5,26,68,113,64,0,
%T 2,5,32,100,241,320,128,0,2,5,38,133,418,844,906,256,0,2,5,44,166,636,
%U 1692,2966,2565,512,0,2,5,50,199,891,2856,6932,10413,7262,1024,0,2,5,56,232
%N T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 1 or less, starting with 0
%C Table starts
%C ....0.....0......0.......0.......0........0........0........0........0
%C ....2.....2......2.......2.......2........2........2........2........2
%C ....4.....5......5.......5.......5........5........5........5........5
%C ....8....14.....20......26......32.......38.......44.......50.......56
%C ...16....40.....68.....100.....133......166......199......232......265
%C ...32...113....241.....418.....636......891.....1182.....1509.....1872
%C ...64...320....844....1692....2856.....4326.....6102.....8184....10572
%C ..128...906...2966....6932...13169....21985....33710....48674....67202
%C ..256..2565..10413...28288...60120...109567...180382...276317...401105
%C ..512..7262..36568..115604..275632...551027...980490..1606710..2476200
%C .1024.20560.128408..472188.1261451..2759757..5289023..9228224.15012528
%C .2048.58209.450913.1929012.5777107.13846871.28634131.53334307.91896348
%H R. H. Hardin, <a href="/A221683/b221683.txt">Table of n, a(n) for n = 1..2080</a>
%F Empirical for column k:
%F k=1: a(n) = 2*a(n-1) for n>2
%F k=2: a(n) = 2*a(n-1) +2*a(n-2) +a(n-3)
%F k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
%F k=4: a(n) = 3*a(n-1) +4*a(n-2) +6*a(n-4) +4*a(n-5) +4*a(n-6)
%F k=5: a(n) = 4*a(n-1) +3*a(n-2) -6*a(n-3) +19*a(n-4) +5*a(n-5) +a(n-6)
%F k=6: a(n) = 4*a(n-1) +5*a(n-2) -7*a(n-3) +33*a(n-4) +17*a(n-5) +24*a(n-6) -5*a(n-7) +2*a(n-8)
%F k=7: a(n) = 5*a(n-1) +3*a(n-2) -16*a(n-3) +65*a(n-4) -14*a(n-5) +23*a(n-6) +2*a(n-7) +8*a(n-8)
%F Empirical for row n:
%F n=1: a(n) = 0
%F n=2: a(n) = 2
%F n=3: a(n) = 5 for n>1
%F n=4: a(n) = 6*n + 2
%F n=5: a(n) = 33*n - 32 for n>3
%F n=6: a(n) = 18*n^2 + 57*n - 99 for n>4
%F n=7: a(n) = 153*n^2 - 213*n + 96 for n>3
%e Some solutions for n=6 k=4
%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e ..1....0....1....1....1....1....0....0....0....0....1....0....0....0....0....1
%e ..2....0....0....0....4....1....4....4....0....2....1....1....1....0....0....2
%e ..1....2....3....1....3....2....3....3....1....3....3....1....2....0....1....3
%e ..1....1....4....2....2....1....2....3....4....2....3....1....1....0....1....0
%e ..0....2....4....2....1....0....3....4....4....2....4....2....1....1....2....1
%Y Row 4 is A016933
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_ Jan 22 2013
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