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A334016 Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess queen from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves.

%I A334016 #22 Feb 21 2021 02:08:41
%S A334016 1,1,1,2,4,6,4,10,21,35,8,25,65,139,237,16,60,179,451,978,1684,32,140,
%T A334016 470,1337,3339,7239,12557,64,320,1189,3725,10325,25559,55423,96605,
%U A334016 128,720,2926,9958,30018,81716,200922,435550,761938,256,1600,7048,25802,83518
%N A334016 Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess queen from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves.
%H A334016 Peter Kagey, <a href="/A334016/b334016.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals)
%H A334016 Peter Kagey, <a href="/A334016/a334016_1.png">Parity bitmap of first 2048 rows and 1024 columns.</a> (Even and odd entries and represented by black and white pixels respectively.)
%F A334016 T(n,k) = Sum_{i=1..k-1} T(n+i, k-i) + Sum_{i=1..min(n,k)-1} T(n-i, k-i) + Sum_{i=1..n-1} T(n-i, k).
%e A334016 Table begins:
%e A334016 n\k|   1    2     3      4       5        6         7          8
%e A334016 ---+------------------------------------------------------------
%e A334016   1|   1    1     6     35     237     1684     12557      96605
%e A334016   2|   1    4    21    139     978     7239     55423     435550
%e A334016   3|   2   10    65    451    3339    25559    200922    1611624
%e A334016   4|   4   25   179   1337   10325    81716    658918    5394051
%e A334016   5|   8   60   470   3725   30018   245220   2027447   16935981
%e A334016   6|  16  140  1189   9958   83518   703635   5961973   50811786
%e A334016   7|  32  320  2926  25802  224831  1951587  16938814  147261146
%e A334016   8|  64  720  7048  65241  589701  5269220  46826316  415175289
%e A334016 For example, the T(2,2) = 4 valid sequences of moves from (1,1) to (2,2) are:
%e A334016 (1,1) -> (2,1) -> (1,2) -> (2,2),
%e A334016 (1,1) -> (2,1) -> (3,1) -> (2,2),
%e A334016 (1,1) -> (2,2), and
%e A334016 (1,1) -> (3,1) -> (2,2).
%Y A334016 Cf. A035002 (up, right), A059450 (right, up-left), A132439 (up, right, up-right), A279212 (up, right, up-right, up-left), A334017 (up, right, up-left).
%Y A334016 A071945 is the analog for king moves. For both king and queen moves, A094727 is the length of the longest sequence of moves.
%K A334016 nonn,tabl
%O A334016 1,4
%A A334016 _Peter Kagey_, Apr 12 2020