login
Table read by antidiagonals: T(n,k) = number of restricted knight's walks from (n,k) (n >= 0, k >= 0).
0

%I #25 Nov 25 2015 04:17:51

%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,3,2,3,1,1,1,1,3,4,4,3,

%T 1,1,1,1,5,5,6,5,5,1,1,1,1,6,7,10,10,7,6,1,1,1,1,8,11,16,14,16,11,8,1,

%U 1,1,1,12,17,22,27,27,22,17,12,1,1,1,1,18,23,39,44,44,44,39,23,18,1,1,1,1

%N Table read by antidiagonals: T(n,k) = number of restricted knight's walks from (n,k) (n >= 0, k >= 0).

%C Moves are restricted to be (-2,1) or (1,-2), and end when either coordinate is < 2.

%H M. Bousquet-Mélou and M. Petkovsek, <a href="http://www.imfm.si/preprinti/PDF/00687.pdf">Linear recurrences with constant coefficients: the multivariate case</a>, Preprint 687, Volume 38 (2000), Preprint Series of the Institute of Mathematics, Physics and Mechanics of Ljubljana.

%H M. Bousquet-Mélou and M. Petkovsek, <a href="http://arXiv.org/abs/math.CO/0211432">Walks confined in a quadrant are not always D-finite</a>, arXiv:math/0211432 [math.CO], 2002.

%e The table starts:

%e 1 1 1 1 1 1

%e 1 1 1 1 1 1

%e 1 1 2 2 3 3

%e 1 1 2 2 4 5

%e 1 1 3 4 6 10

%e 1 1 3 5 10 14

%p T:=proc(n,k) if n=0 or n=1 or k=0 or k=1 then 1 else T(n+1,k-2)+T(n-2,k+1) fi end: seq(seq(T(n,s-n),n=0..s),s=0..15);

%t t[n_, k_] := t[n, k] = If[ n == 0 || n == 1 || k == 0 || k == 1, 1, t[n + 1, k - 2] + t[n - 2, k + 1]]; Table[t[n, s - n], {s, 0, 13}, {n, 0, s}] // Flatten (* _Jean-François Alcover_, Jan 29 2013, translated from Maple *)

%K nonn,tabl,easy,nice

%O 0,13

%A _N. J. A. Sloane_, Nov 05 2000

%E More terms from _Emeric Deutsch_, May 23 2004

%E Definition clarified by _Franklin T. Adams-Watters_, Jun 12 2014