%I #4 Jan 29 2016 12:48:11
%S 3,7,5,13,17,9,21,43,42,17,31,89,143,106,33,43,161,378,479,273,65,57,
%T 265,837,1610,1616,717,129,73,407,1634,4357,6877,5492,1918,257,91,593,
%U 2907,10082,22710,29461,18804,5218,513,111,829,4818,20771,62249,118530,126591
%N T(n,k)=Number of length-(n+1) 0..k arrays with new repeated values introduced in sequential order starting with zero.
%C Table starts
%C ....3.....7.....13.......21.......31........43.........57..........73
%C ....5....17.....43.......89......161.......265........407.........593
%C ....9....42....143......378......837......1634.......2907........4818
%C ...17...106....479.....1610.....4357.....10082......20771.......39154
%C ...33...273...1616.....6877....22710.....62249.....148468......318261
%C ...65...717...5492....29461...118530....384605....1061632.....2587557
%C ..129..1918..18804...126591...619490...2377935....7594224....21042479
%C ..257..5218..64869...545627..3242265..14712729...54345509...171161319
%C ..513.14413.225483..2359152.16993552..91096234..389060724..1392571084
%C .1025.40349.789747.10233188.89197862.564452368.2786424182.11332701236
%H R. H. Hardin, <a href="/A268261/b268261.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical for column k:
%F k=1: a(n) = 3*a(n-1) -2*a(n-2)
%F k=2: a(n) = 7*a(n-1) -15*a(n-2) +7*a(n-3) +6*a(n-4)
%F k=3: a(n) = 13*a(n-1) -60*a(n-2) +105*a(n-3) -11*a(n-4) -94*a(n-5) -24*a(n-6)
%F k=4: [order 8]
%F k=5: [order 10]
%F k=6: [order 12]
%F k=7: [order 14]
%F Empirical for row n:
%F n=1: a(n) = n^2 + n + 1
%F n=2: a(n) = n^3 + n^2 + 2*n + 1
%F n=3: a(n) = n^4 + n^3 + 3*n^2 + 2*n + 2
%F n=4: a(n) = n^5 + n^4 + 4*n^3 + 3*n^2 + 6*n + 2
%F n=5: a(n) = n^6 + n^5 + 5*n^4 + 4*n^3 + 12*n^2 + 6*n + 5 for n>1
%F n=6: a(n) = n^7 + n^6 + 6*n^5 + 5*n^4 + 20*n^3 + 12*n^2 + 20*n + 5 for n>1
%F n=7: a(n) = n^8 + n^7 + 7*n^6 + 6*n^5 + 30*n^4 + 20*n^3 + 50*n^2 + 20*n + 15 for n>2
%e Some solutions for n=5 k=4
%e ..0....4....1....2....0....3....2....1....0....1....4....1....1....1....3....2
%e ..0....3....4....4....0....2....4....0....3....4....0....3....0....2....4....0
%e ..4....4....0....0....4....3....3....0....2....1....4....4....4....3....0....0
%e ..3....0....2....0....0....4....4....4....0....4....1....0....3....2....3....4
%e ..1....0....3....1....2....0....1....3....3....0....0....3....1....3....0....3
%e ..3....3....2....4....0....0....4....0....2....3....3....1....3....0....4....0
%Y Column 1 is A000051.
%Y Row 1 is A002061(n+1).
%Y Row 2 is A100705.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Jan 29 2016
|