A246479 T(n,k)=Number of length n+3 0..k arrays with no pair in any consecutive four terms totalling exactly k.

2, 10, 2, 60, 14, 2, 172, 132, 20, 2, 462, 484, 292, 28, 2, 966, 1734, 1376, 644, 38, 2, 1880, 4386, 6534, 3904, 1420, 52, 2, 3256, 10376, 20004, 24582, 11020, 3132, 72, 2, 5370, 20840, 57416, 91212, 92478, 31104, 6908, 100, 2, 8290, 39690, 133664, 317576

Offset: 1

Author

R. H. Hardin, Aug 27 2014

Keywords

• nonn
• tabl

Comments

Table starts

.2..10....60.....172......462.......966.......1880........3256.........5370

.2..14...132.....484.....1734......4386......10376.......20840........39690

.2..20...292....1376.....6534.....20004......57416......133664.......293770

.2..28...644....3904....24582.....91212.....317576......857248......2174090

.2..38..1420...11020....92478....415650....1756472.....5497304.....16089370

.2..52..3132...31104...347934...1893780....9714968....35251360....119069850

.2..72..6908...87888..1309038...8628792...53733080...226048032....881180090

.2.100.15236..248568..4924998..39320988..297195272..1449551536...6521200010

.2.138.33604..702724.18529350.179184654.1643773832..9295405128..48260338570

.2.190.74116.1985932.69713094.816514170.9091640072.59607621016.357152100490

Examples

			Some solutions for n=5 k=4
..0....3....0....2....2....2....3....2....2....0....4....4....0....0....1....0
..2....4....0....3....0....4....4....1....0....1....3....3....0....0....0....2
..1....3....2....0....3....4....2....0....0....0....3....3....0....3....2....1
..0....4....0....3....3....3....4....0....0....1....3....4....1....2....1....1
..0....4....1....0....3....4....4....0....1....1....3....3....0....3....1....4
..0....3....0....0....4....3....4....1....0....0....3....4....1....4....4....4
..1....3....1....2....3....3....4....0....0....1....2....4....0....3....1....1
..0....3....1....3....3....4....4....1....1....2....3....3....1....4....1....1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..9999

Programs

Maple

G:= proc(m,k) # first m terms in column k
  local q,r,s,S,nS,M,u,v,V,i;
  S:= remove(t -> t[1]+t[2]=k or t[1]+t[3]=k or t[2]+t[3]=k, [seq(seq(seq([q,r,s],s=0..k),r=0..k),q=0..k)]);
nS:= nops(S);
M:= Matrix(nS,nS,(i,j) -> `if`(S[i][2..3] = S[j][1..2] and S[i][1] + S[j][3] <> k, 1, 0));
  u:= Vector[column](nS,1); v:= u;
  V:= Vector(m);
  for i from 1 to m  do
    v:= M . v;
    V[i]:= u^%T . v
  od;
  V
end proc:
R:= Matrix(10,20):
interface(rtablesize=[10,20]):
for j from 1 to 20 do R[.., j] := G(10, j) od:
R; # Robert Israel, Nov 10 2024

Formula

Empirical for column k:

k=1: a(n) = a(n-1)

k=2: a(n) = a(n-1) +a(n-4)

k=3: a(n) = 2*a(n-1) +a(n-3)

k=4: a(n) = 2*a(n-1) +a(n-3) +14*a(n-4) +3*a(n-5) +6*a(n-6) +a(n-8) +a(n-9)

k=5: a(n) = 3*a(n-1) +2*a(n-2) +3*a(n-3) +a(n-4)

k=6: [order 10]

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

k=8: [order 10]

k=9: a(n) = 7*a(n-1) +2*a(n-2) +7*a(n-3) +a(n-4)

From Robert Israel, Nov 10 2024: (Start)

It appears that for k >= 5 odd, the recurrence for column k is

a(n) = (k - 2)*a(n-1) + 2*a(n-2) + (k - 2)*a(n-3) + a(n-4)

and that for k >= 6 even, the recurrence for column k is

a(n) = (k - 3)*a(n-1) + 2*a(n-2) + (k-3)*a(n-3) + (k^3 - 6*k^2 + 15*k - 13)*a(n-4) + (3*k^2 - 11*k + 13)*a(n-5) + (k^3 - 7*k^2 + 19*k - 19)*a(n-6) + (k^2 - 4*k + 6)*a(n-7) + a(n-8) + (k - 2)*a(n-9) + a(n-10). (End)

Empirical for row n:

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

n=2: a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9)

n=3: [order 11]

n=4: [order 13]

n=5: [order 15]

n=6: [order 17]

n=7: [order 19]