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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A278259 A293473 A353512

Adjacent sequences:  A334013 A334014 A334015 * A334017 A334018 A334019

KEYWORD

nonn,tabl

AUTHOR

Peter Kagey, Apr 12 2020

STATUS

approved

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Last modified June 26 14:05 EDT 2022. Contains 354884 sequences. (Running on oeis4.)