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Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).
3

%I #15 May 14 2020 05:35:43

%S 1,13,818,64324,5592968,515092048,49239783968,4831678931008,

%T 483371425775744,49083260519243008,5043379069021557248,

%U 523221884090930480128,54715789513061864081408,5760456190025868833542144,609948004367577499751948288,64905519628343663567453569024

%N Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).

%H Alois P. Heinz, <a href="/A316673/b316673.txt">Table of n, a(n) for n = 0..487</a>

%F Recurrence: see Maple program.

%F a(n) = A126086(n) * ceiling(2^(n-1)) = A126086(n) * A011782(n).

%F a(n) ~ sqrt((6 + 5*2^(1/3) + 4*2^(2/3))/6) * (24*2^(2/3) + 30*2^(1/3) + 38)^n / (4*Pi*n). - _Vaclav Kotesovec_, May 14 2020

%p a:= proc(n) option remember; `if`(n<4, [1, 13, 818, 64324][n+1],

%p (2*(3*n-2)*(57*n^2-95*n+25)*a(n-1)-4*(9*n^3-30*n^2+29*n-6)*

%p a(n-2)+8*(3*n-1)*(n-2)^2*a(n-3))/(n^2*(3*n-4)))

%p end:

%p seq(a(n), n=0..20);

%t a[n_] := a[n] = If[n < 4, {1, 13, 818, 64324}[[n+1]], (2(3n-2)(57n^2- 95n+25) a[n-1] - 4(9n^3-30n^2+29n-6) a[n-2] + 8(3n-1)(n-2)^2 a[n-3]) / (n^2 (3n-4))];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, May 14 2020, after Maple *)

%Y Column k=3 of A316674.

%Y Cf. A052141, A126086.

%K nonn,walk

%O 0,2

%A _Alois P. Heinz_, Jul 10 2018