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A268261 T(n,k)=Number of length-(n+1) 0..k arrays with new repeated values introduced in sequential order starting with zero. 13

%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

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)