A393390 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.

1, 2, 1, 6, 4, 1, 19, 28, 4, 1, 63, 234, 28, 4, 1, 216, 2162, 234, 28, 4, 1, 760, 20972, 2162, 234, 28, 4, 1, 2725, 209608, 21272, 2162, 234, 28, 4, 1, 9910, 2135572, 218740, 21272, 2162, 234, 28, 4, 1, 36446, 22049959, 2323730, 218740, 21272, 2162, 234, 28, 4, 1

Offset

1,2

Comments

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

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.

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..120 (first 15 antidiagonals)

Donald E. Knuth, Leaper graphs, The Mathematical Gazette 78 (1994), Issue 483, 274-297; arXiv:math/9411240 [math.CO], 1994.

Formula

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).

Example

Array begins:
    |       | k
  n | (r,s) | 1  2  3   4    5     6      7       8        9        10         11
  --+-------+--------------------------------------------------------------------
  1 | (1,0) | 1  2  6  19   63   216    760    2725     9910     36446     135268
  2 | (2,1) | 1  4 28 234 2162 20972 209608 2135572 22049959 229939414 2416816416
  3 | (3,2) | 1  4 28 234 2162 21272 218740 2323730 25314097 281299736 3176220308
  4 | (4,1) | 1  4 28 234 2162 21272 218740 2323730 25314097 281333756 3177896808
  5 | (4,3) | 1  4 28 234 2162 21272 218740 2323730 25314097 281345096 3178474308
  6 | (5,2) | 1  4 28 234 2162 21272 218740 2323730 25314097 281345096 3178474308
  7 | (5,4) | 1  4 28 234 2162 21272 218740 2323730 25314097 281345096 3178474308
.
Differences A151830(k)-T(n,k) for n >= 2:
  n\k| 1  2  3  4  5   6    7      8       9       10        11          12
  ---+---------------------------------------------------------------------
   2 | 0  0  0  0  0 300 9132 188158 3264138 51405682 761657892 10830860326
   3 | 0  0  0  0  0   0    0      0       0    45360   2254000    67894904
   4 | 0  0  0  0  0   0    0      0       0    11340    577500    18073440
   5 | 0  0  0  0  0   0    0      0       0        0         0           0
   6 | 0  0  0  0  0   0    0      0       0        0         0           0
   7 | 0  0  0  0  0   0    0      0       0        0         0           0

Crossrefs

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

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

Sequence in context: A073387 A259099 A125693 * A094527 A054335 A110681

Adjacent sequences: A393387 A393388 A393389 * A393391 A393392 A393393

Keyword

nonn,tabl

Author

Pontus von Brömssen, Feb 14 2026

Status

approved