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

1,1

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

Links

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

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]

Example

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

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

Crossrefs

Sequence in context: A347096 A346239 A188635 * A359694 A171659 A060466

Adjacent sequences: A246476 A246477 A246478 * A246480 A246481 A246482

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 27 2014

Status

approved