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%I #4 Jan 20 2013 04:57:52
%S 0,0,2,0,5,2,0,10,9,4,0,17,26,25,6,0,26,59,100,57,10,0,37,114,289,342,
%T 141,16,0,50,197,676,1293,1210,345,26,0,65,314,1369,3734,5913,4240,
%U 853,42,0,82,471,2500,8991,20944,26911,14898,2097,68,0,101,674,4225,19014
%N T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1
%C Table starts
%C ...0.....0.......0........0.........0..........0...........0............0
%C ...2.....5......10.......17........26.........37..........50...........65
%C ...2.....9......26.......59.......114........197.........314..........471
%C ...4....25.....100......289.......676.......1369........2500.........4225
%C ...6....57.....342.....1293......3734.......8991.......19014........36497
%C ..10...141....1210.....5913.....20944......59705......145800.......317233
%C ..16...345....4240....26911....117104.....395641.....1116400......2754635
%C ..26...853...14898...122621....655198....2622817.....8550512.....23923281
%C ..42..2097...52306...558547...3665306...17385993....65485386....207761745
%C ..68..5149..183684..2544357..20505052..115249117...501533796...1804315029
%C .110.12633..645006.11590169.114711980..763966685..3841097940..15669633131
%C .178.31013.2264978.52796369.641737294.5064207645.29417832750.136083460405
%H R. H. Hardin, <a href="/A221573/b221573.txt">Table of n, a(n) for n = 1..2080</a>
%F Empirical for column k:
%F k=1: a(n) = a(n-1) +a(n-2)
%F k=2: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-4)
%F k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
%F k=4: a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5) for n>6
%F k=5: a(n) = 5*a(n-1) +3*a(n-2) +9*a(n-4) +6*a(n-5) +3*a(n-6)
%F k=6: a(n) = 7*a(n-1) -4*a(n-2) +6*a(n-3) +26*a(n-4) +10*a(n-5) +16*a(n-6) +12*a(n-8)
%F k=7: a(n) = 7*a(n-1) +4*a(n-2) +5*a(n-3) +20*a(n-4) +20*a(n-5) +23*a(n-6) -6*a(n-7) +3*a(n-8)
%F Empirical for row n:
%F n=2: a(n) = 1*n^2 + 1
%F n=3: a(n) = 1*n^3 - 1*n^2 + 3*n - 1
%F n=4: a(n) = 1*n^4 + 2*n^2 + 1
%F n=5: a(n) = 1*n^5 + 1*n^4 - 2*n^3 + 12*n^2 - 15*n + 9 for n>2
%F n=6: a(n) = 1*n^6 + 2*n^5 - 5*n^4 + 24*n^3 - 41*n^2 + 50*n - 31 for n>3
%F n=7: a(n) = 1*n^7 + 3*n^6 - 7*n^5 + 29*n^4 - 41*n^3 + 45*n^2 - 33*n + 19 for n>2
%e Some solutions for n=6 k=4
%e ..2....2....3....1....0....0....2....1....2....2....3....1....4....3....1....4
%e ..0....0....0....1....4....0....2....3....0....4....0....3....4....3....3....4
%e ..1....1....4....3....4....2....0....4....0....0....4....2....3....0....1....2
%e ..3....1....2....1....3....3....2....4....0....3....3....2....1....3....0....0
%e ..3....4....1....1....0....0....0....3....1....2....1....4....0....3....3....0
%e ..0....1....4....1....3....0....4....0....3....4....3....4....2....3....0....0
%Y Column 1 is A006355
%Y Row 2 is A002522
%Y Row 4 is A082044
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_ Jan 20 2013