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A358481 a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point. 1

%I #8 Dec 21 2022 12:03:07

%S 30,6218,2658432,1054788750,552306591900,269380692717696,

%T 155175092086118400,83798883891736779150,50885239237727996887500,

%U 29198209396114625497699068,18332853214682572877389897728,10951674446687597386319569942656,7036938452279110885561897815723264,4325988198220149508865311059521280000

%N a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.

%C Alternatively a(n) is the number of ways two ants can interchange their positions starting simultaneously from two opposite corners and moving along shortest grid paths at same speed in an n X n X n grid without meeting other one.

%F a(n) = A268553(n) - A357760(n).

%e When n=2 number of ways to move between two opposite corners are given by 6!/(2!*2!*2!) and number of such pairs are given by (6!/(2!*2!*2!))^2. This total number of pairs are given by A268553(2)=8100.

%e Number of pairs which have the middle point of their paths as a common point are A357760(2)=1782.

%e Therefore number of pairs without having middle point on their paths as a common point are 8100-1782=6218

%Y Cf. A268553, A357760.

%K nonn

%O 1,1

%A _Janaka Rodrigo_, Nov 18 2022

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Last modified July 22 23:31 EDT 2024. Contains 374544 sequences. (Running on oeis4.)