A241619 - OEIS

%I #8 Sep 04 2019 10:28:48

%S 4,10,6,20,20,9,35,50,40,13,56,105,125,76,19,84,196,315,295,147,28,

%T 120,336,686,889,711,287,41,165,540,1344,2254,2567,1730,556,60,220,

%U 825,2430,5040,7586,7483,4175,1077,88,286,1210,4125,10242,19374,25774,21631,10077

%N T(n,k)=Number of length n+2 0..k arrays with no consecutive three elements summing to more than k

%C Table starts

%C ...4...10....20.....35......56.......84......120.......165.......220........286

%C ...6...20....50....105.....196......336......540.......825......1210.......1716

%C ...9...40...125....315.....686.....1344.....2430......4125......6655......10296

%C ..13...76...295....889....2254.....5040....10242.....19305.....34243......57772

%C ..19..147...711...2567....7586....19374....44274.....92697....180829.....332761

%C ..28..287..1730...7483...25774....75180...193194....449295....963886....1934647

%C ..41..556..4175..21631...86828...289248...835812...2159025...5093737...11151140

%C ..60.1077.10077..62547..292621..1113348..3617703..10380183..26932543...64309245

%C ..88.2091.24377.181255..988303..4294574.15692003..50011289.142701909..371651553

%C .129.4057.58928.524877.3335451.16553380.68014233.240772037.755538278.2146210209

%H R. H. Hardin, <a href="/A241619/b241619.txt">Table of n, a(n) for n = 1..10011</a>

%F Empirical for column k, apparently a recurrence of order (k+1)*(k+2)/2:

%F k=1: a(n) = a(n-1) +a(n-3)

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

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

%F k=4: [order 15]

%F k=5: [order 21]

%F k=6: [order 28]

%F k=7: [order 36]

%F k=8: [order 45]

%F k=9: [order 55]

%F k=10: [order 66]

%F k=11: [order 78]

%F k=12: [order 91]

%F Empirical for row n, apparently a polynomial of degree n+2:

%F n=1: a(n) = (1/6)*n^3 + 1*n^2 + (11/6)*n + 1

%F n=2: a(n) = (1/12)*n^4 + (2/3)*n^3 + (23/12)*n^2 + (7/3)*n + 1

%F n=3: a(n) = (1/24)*n^5 + (5/12)*n^4 + (13/8)*n^3 + (37/12)*n^2 + (17/6)*n + 1

%F n=4: [polynomial of degree 6]

%F n=5: [polynomial of degree 7]

%F n=6: [polynomial of degree 8]

%F n=7: [polynomial of degree 9]

%F From _Robert Israel_, Sep 04 2019: (Start)

%F Column k satisfies a recurrence of order (k+1)*(k+2)/2, since a(n)=e^T T^n e where T is a (k+1)*(k+2)/2 matrix and e the vector of all 1's (see proofs at A241615 and A241618).

%F Row n is the Ehrhart polynomial of degree n+2 corresponding to the polytope {(x(1),...,x(n+2)): all x(i)>=0, x(i)+x(i+1)+x(i+2)<=1 for i=1..n}, whose vertices have all entries in {0,1}. (End)

%e Some solutions for n=5 k=4

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

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

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

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

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

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

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

%p for m from 1 to 12 do

%p   r:= [seq(seq([i,j],j=0..m-i),i=0..m)];

%p   T[m]:= Matrix((m+1)*(m+2)/2,(m+1)*(m+2)/2, proc(i, j) if r[i][1]=r[j][2] and r[i][1]+r[i][2]+r[j][1]<=m then 1 else 0 fi end proc):

%p   U[m,0]:= Vector((m+1)*(m+2)/2,1);

%p od:

%p R:= NULL:

%p for i from 2 to 12 do

%p   for j from 1 to i-1 do

%p     U[i-j,j]:= T[i-j] . U[i-j,j-1];

%p     R:= R, convert(U[i-j,j],`+`)

%p od od:

%p R; # _Robert Israel_, Sep 04 2019

%Y Column 1 is A000930(n+4)

%Y Row 1 is A000292(n+1)

%Y Row 2 is A002415(n+2)

%Y Row 3 is A006414

%Y Row 4 is A114244

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Apr 26 2014