[seqfan] Re: knight's distnaces

[Richard J. Mathar]

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