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
Examples
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)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (7,2).
Programs
-
Maple
a:= n-> (<<0|1>, <2|7>>^n. <<1, 6>>)[1, 1]: seq(a(n), n=0..25);
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.