A227665 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component by 1 such that for each point (p_1,p_2,p_3) we have abs(p_{i}-p_{i+1}) <= 1.

1, 6, 44, 320, 2328, 16936, 123208, 896328, 6520712, 47437640, 345104904, 2510609608, 18264477064, 132872558664, 966636864776, 7032203170760, 51158695924872, 372175277815624, 2707544336559112, 19697160911545032, 143295215053933448, 1042460827200624200

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,2).

Formula

G.f.: (x-1)/(2*x^2+7*x-1).

a(n) = 7*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(2)=6.

Example

a(1) = 3! = 3*2*1 = 6:
            (0,1,1) - (0,0,1)
          /         X         \
  (1,1,1) - (1,0,1)   (0,1,0) - (0,0,0)
          \         X         /
            (1,1,0) - (1,0,0)

Maple

a:= n-> (<<0|1>, <2|7>>^n. <<1, 6>>)[1, 1]:
seq(a(n), n=0..25);

Crossrefs

Column k=3 of A227655.

Cf. A000142.

Sequence in context: A156002 A091163 A189800 * A102591 A114935 A115969

Adjacent sequences: A227662 A227663 A227664 * A227666 A227667 A227668

Keyword

nonn,easy

Author

Alois P. Heinz, Jul 19 2013

Status

approved