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.

1, 1, 1, 2, 4, 6, 4, 10, 21, 35, 8, 25, 65, 139, 237, 16, 60, 179, 451, 978, 1684, 32, 140, 470, 1337, 3339, 7239, 12557, 64, 320, 1189, 3725, 10325, 25559, 55423, 96605, 128, 720, 2926, 9958, 30018, 81716, 200922, 435550, 761938, 256, 1600, 7048, 25802, 83518

Offset

1,4

Links

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals)

Peter Kagey, Parity bitmap of first 2048 rows and 1024 columns. (Even and odd entries and represented by black and white pixels respectively.)

Formula

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).

Example

Table begins:
n\k|   1    2     3      4       5        6         7          8
---+------------------------------------------------------------
  1|   1    1     6     35     237     1684     12557      96605
  2|   1    4    21    139     978     7239     55423     435550
  3|   2   10    65    451    3339    25559    200922    1611624
  4|   4   25   179   1337   10325    81716    658918    5394051
  5|   8   60   470   3725   30018   245220   2027447   16935981
  6|  16  140  1189   9958   83518   703635   5961973   50811786
  7|  32  320  2926  25802  224831  1951587  16938814  147261146
  8|  64  720  7048  65241  589701  5269220  46826316  415175289
For example, the T(2,2) = 4 valid sequences of moves from (1,1) to (2,2) are:
(1,1) -> (2,1) -> (1,2) -> (2,2),
(1,1) -> (2,1) -> (3,1) -> (2,2),
(1,1) -> (2,2), and
(1,1) -> (3,1) -> (2,2).

Crossrefs

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).

A071945 is the analog for king moves. For both king and queen moves, A094727 is the length of the longest sequence of moves.

Sequence in context: A014428 A180935 A151886 * A371241 A278259 A293473

Adjacent sequences: A334013 A334014 A334015 * A334017 A334018 A334019

Keyword

nonn,tabl

Author

Peter Kagey, Apr 12 2020

Status

approved