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A357603 a(n) is the number of different pairs of shortest paths in an n X n lattice going between opposite corners in opposite directions and not meeting at their middle point. 1
0, 2, 18, 236, 3090, 42252, 589932, 8383608, 120720402, 1756863020, 25789460268, 381298472568, 5671808350572, 84807208655288, 1273785187835640, 19207311526394736, 290631247129611282, 4411188317020786668, 67137528193253129484, 1024357917198436543800 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Equivalently, a(n) is the number of different ways to interchange the positions of two men standing at opposite corners of an n X n lattice without meeting each other.
The two men start to move simultaneously at the same constant speed; one always moves to the right or upward, the other always moves to the left or downward.
All terms are even.
LINKS
FORMULA
a(n) = A002894(n) - A005260(n).
EXAMPLE
Let the lattice points of a lattice of size 2 X 2 be labeled 1,2,3,4,5,6,7,8,9, and let men A and B start at points 1 and 9, respectively.
man B
starts
7---8---9 <-- here
| | |
man A 4---5---6
starts | | |
here --> 1---2---3
.
The lattice paths available for A are 14789, 14589, 14569, 12589, 12569, 12369 and those available for B are 98741, 98541, 98521, 96541, 96521, 96321.
A002894(2) = 36 is the number of different ways to exchange positions, that is, 6 X 6 or (C(4,2))^2.
The different ways they can meet halfway on their paths are as follows:
If A selects 14789, B must select 98741. If A selects one of 14589, 14569, 12589, 12569, B must select one of 98541, 98521, 96541, 96521. If A selects 12369, B must select 96321.
Therefore the total number of choices available is 1 + 4*4 + 1 = 18 and this is given by A005260(2) = (C(2,0))^4 + (C(2,1))^4 + (C(2,2))^4 = 18.
Therefore the total number of such different pairs is a(2) = 36 - 18 = 18.
CROSSREFS
Sequence in context: A260332 A254999 A364167 * A024486 A052635 A366001
KEYWORD
nonn,easy
AUTHOR
Janaka Rodrigo, Oct 05 2022
STATUS
approved

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Last modified February 21 13:52 EST 2024. Contains 370235 sequences. (Running on oeis4.)