# [seqfan] Re: knight's distnaces

Richard J. Mathar mathar at mpia-hd.mpg.de
Thu Jan 4 19:40:57 CET 2018

```Some partial answers to
http://list.seqfan.eu/pipermail/seqfan/2017-December/018205.html

The number of steps of the (1,2)-leaper to reach (n,0) gives A018837

The number of steps of the (1,2)-leaper to reach (n,n) gives A018838

The number of steps of the (2,3)-leaper to reach (n,0) gives  A018840.

The number of steps of the (1,3)-leaper to reach (n,0) gives
essentially a diluted A018838, where -1 for odd n indicates that n is not
reachable:
0,-1,2,-1,4,-1,2,-1,4,-1,4,-1,4,-1,6,-1,6,-1,6,-1,8,-1,8,-1,8,-1,10,
-1,10,-1,10,-1,12,-1,12,-1,12,-1,14,-1,14,-1,14,-1,16,-1,16,-1,16,
-1,18,-1,18,-1,18,-1,20,-1,20,-1,20,-1,....

The number of steps of the (1,3)-leaper to reach (n,n) gives
apparently another interpretation of A018837.

The number of steps of the (1,4)-leaper to reach (n,0) gives
0,5,2,7,4,7,4,5,2,5,4,7,6,7,6,5,4,5,6,7,8,9,8,7,6,7,8,9,10,11,10,9,8,9,
10,11,12,13,12,11,10,11,12,13,14,15,14,13,12,13,14,15,16,17,16,15,14,15,
16,17,18,19,18,17,16,17,18,19,20,21,20,19,18,19,20,21,...

The number of steps of the (1,5)-leaper to reach (n,0) gives
0,-1,2,-1,4,-1,6,-1,4,-1,2,-1,4,-1,6,-1,6,-1,6,-1,4,-1,6,-1,6,-1,6,
-1,8,-1,6,-1,8,-1,8,-1,8,-1,10,-1,8,-1,10,-1,10,-1,10,...

The number of steps of the (1,5)-leaper to reach (n,n) gives
apparently another interpretation of A018840.

The number of steps of the (2,5)-leaper to reach (n,0) gives
0,7,6,9,2,7,4,9,4,7,2,7,6,9,4,7,6,9,6,7,4,7,8,9,6,7,8,11,8,7,6,9,10,
9,8,9,10,11,10,9,8,11,12,...

The number of steps of the (2,5)-leaper to reach (n,n) gives
0,6,8,2,4,10,4,2,8,6,4,6,8,6,4,10,8,6,8,10,8,6,10,10,8,10,10,10,8,10,
12,10,12,12,12,10,12,14,12,14,...

The number of steps of the (3,4)-leaper to reach (n,0) gives
0,7,4,7,6,7,2,7,2,7,6,7,4,7,4,9,4,7,6,7,6,9,6,9,6,7,8,9,8,11,8,9,8,9,
10,11,10,11,10,11,...

The number of steps of the (3,4)-leaper to reach (n,n) gives
0,2,4,6,8,6,4,2,4,6,8,8,8,6,4,6,8,8,8,10,8,6,8,10,8,8,10,10,8,10,12,
10,10,12,12,10,12,...

Coverages:

The number of distinct positions reachable by the (1,2)-leaper in n or less steps
is A018836.

The number of distinct positions reachable by the (2,3)-leaper in n or less steps
is
9, 41, 129, 321, 625, 997, 1413, 1885, 2425, 3033, 3709, 4453, 5265,
6145, 7093, 8109, 9193, 10345, 11565, 12853, 14209,...

The number of distinct positions reachable by the (1,3)-leaper in n or less steps
is (apparently) also A018836.

The number of distinct positions reachable by the (3,4)-leaper in n or less steps
is
9, 41, 129, 321, 681, 1289, 2121, 3081, 4121, 5233, 6445, 7777, 9233,
10813, 12517, 14345, 16297, 18373, 20573, 22897, 25345,...

Independent confirmations are desirable!

Richard
```