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A334017 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 up, right, and diagonal up-left moves.

%I A334017 #19 Feb 21 2021 02:08:49
%S A334017 1,1,2,2,5,10,4,13,33,63,8,32,98,240,454,16,76,269,777,1871,3539,32,
%T A334017 176,702,2295,6420,15314,29008,64,400,1768,6393,19970,54758,129825,
%U A334017 246255,128,896,4336,17088,58342,176971,478662,1129967,2145722,256,1984,10416
%N A334017 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 up, right, and diagonal up-left moves.
%C A334017 First row is A175962.
%H A334017 Peter Kagey, <a href="/A334017/b334017.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals)
%H A334017 Peter Kagey, <a href="/A334017/a334017.png">Parity bitmap for first 1024 rows and columns</a>. (Even and odd entries and represented by black and white pixels respectively.)
%e A334017 Table begins:
%e A334017 n\k|  1   2     3      4       5        6         7          8
%e A334017 ---+----------------------------------------------------------
%e A334017   1|  1   2    10     63     454     3539     29008     246255
%e A334017   2|  1   5    33    240    1871    15314    129825    1129967
%e A334017   3|  2  13    98    777    6420    54758    478662    4266102
%e A334017   4|  4  32   269   2295   19970   176971   1593093   14532881
%e A334017   5|  8  76   702   6393   58342   536080   4965056   46345046
%e A334017   6| 16 176  1768  17088  163041  1550809  14765863  140982374
%e A334017   7| 32 400  4336  44280  440602  4332221  42373370  413689403
%e A334017   8| 64 896 10416 111984 1159580 11771312 118190333 1179448443
%e A334017 For example, the T(2,2) = 5 sequences of permissible queen's moves from (1,1) to (2,2) are:
%e A334017 (1,1) -> (1,2) -> (2,2),
%e A334017 (1,1) -> (2,1) -> (1,2) -> (2,2),
%e A334017 (1,1) -> (2,1) -> (2,2),
%e A334017 (1,1) -> (2,1) -> (3,1) -> (2,2), and
%e A334017 (1,1) -> (3,1) -> (2,2).
%Y A334017 Cf. A175962.
%Y A334017 Cf. A035002 (up, right), A059450 (right, up-left), A132439 (up, right, up-right), A279212 (up, right, up-left), A334016 (right, up-right, up-left).
%Y A334017 A033877 is the analog for king moves. For both king and queen moves, A094727 is the length of the longest sequence of moves.
%K A334017 nonn,tabl
%O A334017 1,3
%A A334017 _Peter Kagey_, Apr 12 2020