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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. 2

%I

%S 1,6,44,320,2328,16936,123208,896328,6520712,47437640,345104904,

%T 2510609608,18264477064,132872558664,966636864776,7032203170760,

%U 51158695924872,372175277815624,2707544336559112,19697160911545032,143295215053933448,1042460827200624200

%N 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.

%H Alois P. Heinz, <a href="/A227665/b227665.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,2).

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

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

%e a(1) = 3! = 3*2*1 = 6:

%e . (0,1,1) - (0,0,1)

%e . / X \

%e (1,1,1) - (1,0,1) (0,1,0) - (0,0,0)

%e . \ X /

%e . (1,1,0) - (1,0,0)

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

%p seq(a(n), n=0..25);

%Y Column k=3 of A227655.

%Y Cf. A000142.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Jul 19 2013

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Last modified September 19 14:36 EDT 2020. Contains 337178 sequences. (Running on oeis4.)