A221596 - OEIS

%I #4 Jan 20 2013 17:18:41

%S 0,0,4,0,7,8,0,10,17,16,0,13,26,49,32,0,16,35,100,139,64,0,19,44,169,

%T 342,393,128,0,22,53,256,651,1210,1113,256,0,25,62,361,1068,2715,4240,

%U 3151,512,0,28,71,484,1593,5082,11011,14898,8921,1024,0,31,80,625,2226,8475

%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

%C Table starts

%C ....0......0.......0........0........0.........0.........0.........0..........0

%C ....4......7......10.......13.......16........19........22........25.........28

%C ....8.....17......26.......35.......44........53........62........71.........80

%C ...16.....49.....100......169......256.......361.......484.......625........784

%C ...32....139.....342......651.....1068......1593......2226......2967.......3816

%C ...64....393....1210.....2715.....5082......8475.....13056.....18987......26430

%C ..128...1113....4240....11011....22912.....41401.....67936....103975.....150976

%C ..256...3151...14898....45099...105586....210101....374342....615965.....954572

%C ..512...8921...52306...184063...482204...1047967...2006006...3504371....5714456

%C .1024..25257..183684...752155..2210256...5267759..10894988..20352239...35218688

%C .2048..71507..645006..3072247.10115926..26387005..58789204.116958723..213700742

%C .4096.202449.2264978.12550859.46327024.132384353.318224626.675761541.1307528098

%H R. H. Hardin, <a href="/A221596/b221596.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) for 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) = 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=2: a(n) = 3*n + 1

%F n=3: a(n) = 9*n - 1

%F n=4: a(n) = 9*n^2 + 6*n + 1

%F n=5: a(n) = 54*n^2 - 69*n + 63 for n>2

%F n=6: a(n) = 27*n^3 + 108*n^2 - 252*n + 267 for n>3

%F n=7: a(n) = 243*n^3 - 351*n^2 + 237*n + 127 for n>2

%e Some solutions for n=6 k=4

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

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

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

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

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

%e ..0....2....1....2....1....3....1....2....2....3....2....3....0....1....0....0

%Y Column 3 is A221568

%Y Row 2 is A016777

%Y Row 3 is A017257(n-1)

%Y Row 4 is A016778

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_ Jan 20 2013