A393390 - OEIS

%I #15 Mar 18 2026 23:10:06

%S 1,2,1,6,4,1,19,28,4,1,63,234,28,4,1,216,2162,234,28,4,1,760,20972,

%T 2162,234,28,4,1,2725,209608,21272,2162,234,28,4,1,9910,2135572,

%U 218740,21272,2162,234,28,4,1,36446,22049959,2323730,218740,21272,2162,234,28,4,1

%N Array read by antidiagonals: T(n,k) is the number of fixed (r,s)-polyleapers with k cells, where r = A094192(n-1) and s = A094193(n-1) if n > 1, (r,s) = (1,0) if n = 1; n, k >= 1.

%C An (r,s)-polyleaper is a polyform consisting of cells in Z^2 connected by (r,s)-leaps.

%C For n >= 1, (r,s) runs through all pairs of nonnegative integers, in lexicographic order, satisfying r > s and GCD(r+s,r-s) = 1. The latter condition ensures that the underlying infinite (r,s)-leaper graph is connected.

%H Pontus von Brömssen, <a href="/A393390/b393390.txt">Table of n, a(n) for n = 1..120</a> (first 15 antidiagonals)

%H Donald E. Knuth, <a href="https://doi.org/10.2307/3620202">Leaper graphs</a>, The Mathematical Gazette 78 (1994), Issue 483, 274-297; <a href="https://arxiv.org/abs/math/9411240">arXiv:math/9411240</a> [math.CO], 1994.

%F T(n,k) = A151830(k) if k < 2*(r+s). This is because the (r,s)-leaper graph is locally isomorphic to the 4-dimensional grid graph, in the sense that balls of radius R in the two graphs are isomorphic for R < r+s, realized by the mapping f: Z^4 -> Z^2 given by f(x_1, x_2, x_3, x_4) = x_1*(r,s) + x_2*(r,-s) + x_3*(s,r) + x_4*(s,-r).

%e Array begins:

%e     |       | k

%e   n | (r,s) | 1  2  3   4    5     6      7       8        9        10         11

%e   --+-------+--------------------------------------------------------------------

%e   1 | (1,0) | 1  2  6  19   63   216    760    2725     9910     36446     135268

%e   2 | (2,1) | 1  4 28 234 2162 20972 209608 2135572 22049959 229939414 2416816416

%e   3 | (3,2) | 1  4 28 234 2162 21272 218740 2323730 25314097 281299736 3176220308

%e   4 | (4,1) | 1  4 28 234 2162 21272 218740 2323730 25314097 281333756 3177896808

%e   5 | (4,3) | 1  4 28 234 2162 21272 218740 2323730 25314097 281345096 3178474308

%e   6 | (5,2) | 1  4 28 234 2162 21272 218740 2323730 25314097 281345096 3178474308

%e   7 | (5,4) | 1  4 28 234 2162 21272 218740 2323730 25314097 281345096 3178474308

%e .

%e Differences A151830(k)-T(n,k) for n >= 2:

%e   n\k| 1  2  3  4  5   6    7      8       9       10        11          12

%e   ---+---------------------------------------------------------------------

%e    2 | 0  0  0  0  0 300 9132 188158 3264138 51405682 761657892 10830860326

%e    3 | 0  0  0  0  0   0    0      0       0    45360   2254000    67894904

%e    4 | 0  0  0  0  0   0    0      0       0    11340    577500    18073440

%e    5 | 0  0  0  0  0   0    0      0       0        0         0           0

%e    6 | 0  0  0  0  0   0    0      0       0        0         0           0

%e    7 | 0  0  0  0  0   0    0      0       0        0         0           0

%Y Rows 1-5 are A001168 (polyominoes), A030444 (polyknights), A093877 (polyzebras), A393836 (polygiraffes), A394194 (polyantelopes).

%Y Cf. A094192, A094193, A151830, A393391 (one-sided), A393392 (free), A393393 (coordination sequences for the underlying leaper graphs).

%K nonn,tabl

%O 1,2

%A _Pontus von Brömssen_, Feb 14 2026