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