A217954 - OEIS

%I #10 Sep 02 2013 09:55:09

%S 4,4,16,4,16,64,4,16,50,256,4,16,50,144,1024,4,16,50,130,422,4096,4,

%T 16,50,130,310,1268,16384,4,16,50,130,296,736,3823,65536,4,16,50,130,

%U 296,624,1821,11472,262144,4,16,50,130,296,610,1289,4673,34350,1048576,4,16

%N T(n,k) = number of n-element 0..3 arrays with each element the minimum of k adjacent elements of a random 0..3 array of n+k-1 elements.

%C See A228461 for comments on the definition. - _N. J. A. Sloane_, Sep 02 2013

%C Table starts

%C ........4......4......4.....4.....4.....4.....4.....4.....4.....4.....4.....4

%C .......16.....16.....16....16....16....16....16....16....16....16....16....16

%C .......64.....50.....50....50....50....50....50....50....50....50....50....50

%C ......256....144....130...130...130...130...130...130...130...130...130...130

%C .....1024....422....310...296...296...296...296...296...296...296...296...296

%C .....4096...1268....736...624...610...610...610...610...610...610...610...610

%C ....16384...3823...1821..1289..1177..1163..1163..1163..1163..1163..1163..1163

%C ....65536..11472...4673..2741..2209..2097..2083..2083..2083..2083..2083..2083

%C ...262144..34350..12107..6134..4202..3670..3558..3544..3544..3544..3544..3544

%C ..1048576.102896..31103.14269..8366..6434..5902..5790..5776..5776..5776..5776

%C ..4194304.308419..79039.33577.17569.11666..9734..9202..9090..9076..9076..9076

%C .16777216.924532.199819.78304.38251.22313.16410.14478.13946.13834.13820.13820

%H R. H. Hardin, <a href="/A217954/b217954.txt">Table of n, a(n) for n = 1..1275</a>

%F Empirical for column k:

%F k=2: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7)

%F k=3: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) +5*a(n-4) -4*a(n-5) +6*a(n-6) +4*a(n-7) +2*a(n-9) +a(n-10)

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

%F k=5: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-6) -4*a(n-7) +6*a(n-8) +4*a(n-9) +5*a(n-10) +6*a(n-11) +2*a(n-12) +4*a(n-13) +3*a(n-14) +2*a(n-15) +a(n-16)

%F k=6: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-7) -4*a(n-8) +6*a(n-9) +4*a(n-10) +5*a(n-11) +6*a(n-12) +7*a(n-13) +3*a(n-14) +5*a(n-15) +4*a(n-16) +3*a(n-17) +2*a(n-18) +a(n-19)

%F k=7: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-8) -4*a(n-9) +6*a(n-10) +4*a(n-11) +5*a(n-12) +6*a(n-13) +7*a(n-14) +8*a(n-15) +4*a(n-16) +6*a(n-17) +5*a(n-18) +4*a(n-19) +3*a(n-20) +2*a(n-21) +a(n-22)

%F Diagonal: a(n) = (1/720)*n^6 + (1/48)*n^5 + (23/144)*n^4 + (9/16)*n^3 + (241/180)*n^2 + (11/12)*n + 1

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 2 is A203094(n+1). A217949 is also a column. Cf. A228461, A217883.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, suggestion that the diagonal might be a polynomial from _L. Edson Jeffery_ in the Sequence Fans Mailing List, Oct 15 2012