A232103 Square array read by antidiagonals: T(m,n) = number of ways of drawing a simple loop on an m x n rectangular lattice of dots in such a way that it touches each edge.
1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 39, 106, 39, 1, 1, 97, 582, 582, 97, 1, 1, 237, 2952, 6074, 2952, 237, 1, 1, 575, 14488, 56778, 56778, 14488, 575, 1, 1, 1391, 69982, 510600, 943340, 510600, 69982, 1391, 1, 1, 3361, 335356, 4502836, 15009212, 15009212
Offset: 1
Examples
Array begins: ============================================================== m\n| 1 2 3 4 5 6 7 ---|---------------------------------------------------------- 1 | 1 1 1 1 1 1 1... 2 | 1 5 15 39 97 237 575... 3 | 1 15 106 582 2952 14488 69982... 4 | 1 39 582 6074 56778 510600 4502836... 5 | 1 97 2952 56778 943340 15009212 234411981... 6 | 1 237 14488 510600 15009212 419355340 11509163051... 7 | 1 575 69982 4502836 234411981 11509163051 554485727288... ... - _Andrew Howroyd_, Oct 04 2017 a(3,2) is 15, thus: 1) 2) 3) 4) 5) +-+-+-+ +-+-+-+ + +-+-+ +-+-+-+ +-+-+-+ | | | | | | | | | | + +-+-+ +-+ +-+ +-+ +-+ + + +-+ +-+-+ + | | | | | | | | | | +-+ + + + +-+ + +-+-+ + +-+-+ + + + +-+ 6) 7) 8) 9) 10) +-+-+-+ +-+-+ + +-+-+-+ +-+ + + + +-+ + | | | | | | | | | | + +-+ + +-+ +-+ +-+ + + + +-+-+ +-+ +-+ | | | | | | | | | | | | +-+ +-+ + +-+-+ + +-+-+ +-+-+-+ +-+-+-+ 11) 12) 13) 14) 15) +-+-+ + + + +-+ +-+ +-+ + +-+-+ +-+-+-+ | | | | | | | | | | | | + +-+ +-+-+ + + +-+ + +-+ + + + + + + | | | | | | | | | | +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..325
- Jean-François Alcover, Mathematica program
Formula
T(m, n) = U(m, n) - 2*U(m, n-1) + U(m, n-2) where U(m, n) = V(m, n) - 2*V(m-1, n) + V(m-2, n) and V(m, n) = A231829(m, n). - Andrew Howroyd, Oct 04 2017
Comments