A132439 -id:A132439 - cp's OEIS frontend

A279212 Fill an array by antidiagonals upwards; in the top left cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that can be seen from that cell.

1, 1, 2, 2, 6, 11, 4, 15, 39, 72, 8, 37, 119, 293, 543, 16, 88, 330, 976, 2364, 4403, 32, 204, 870, 2944, 8373, 20072, 37527, 64, 464, 2209, 8334, 26683, 74150, 176609, 331072, 128, 1040, 5454, 22579, 79534, 246035, 673156, 1595909, 2997466, 256, 2304, 13176, 59185, 226106, 762221, 2303159, 6231191, 14721429, 27690124

Offset: 0

N. J. A. Sloane, Dec 24 2016

"That can be seen from" means "that are on the same row, column, diagonal, or antidiagonal as".
Inspired by A279967.
Conjecture: Every column has a finite number of odd entries, and every row and diagonal have an infinite number of odd entries. - Peter Kagey, Mar 28 2020. The conjecture about columns is true, see that attached pdf file from Alec Jones.
The "look" keyword refers to Peter Kagey's bitmap. - N. J. A. Sloane, Mar 29 2020
The number of sequences of queen moves from (1, 1) to (n, k) in the first quadrant moving only up, right, diagonally up-right, or diagonally up-left. - Peter Kagey, Apr 12 2020
Column 0 gives A011782. In the column 1, the only powers of 2 occur at positions A233328(k) with value a(k(k+1)/2 + 1), k >=1 (see A335903). Conjecture: Those are the only multiple occurrences of numbers greater than 1 in this sequence (checked through the first 2000 antidiagonals). - Hartmut F. W. Hoft, Jun 29 2020

			The array begins:
i/j|  0    1    2     3     4      5      6       7       8
-------------------------------------------------------------
0  |  1    2   11    72   543   4403  37527  331072 2997466 ...
1  |  1    6   39   293  2364  20072 176609 1595909 ...
2  |  2   15  119   976  8373  74150 673156 ...
3  |  4   37  330  2944 26683 246035 ...
4  |  8   88  870  8334 79534 ...
5  | 16  204 2209 22579 ...
6  | 32  464 5454 ...
7  | 64 1040 ...
8  |128 ...
  ...
For example, when we get to the antidiagonal that reads 4, 15, 39, ..., the reason for the 39 is that from that cell we can see one cell that has been filled in above it (containing 11), one cell to the northwest (2), two cells to the west (1, 6), and two to the southwest (4, 15), for a total of a(8) = 39.
The next pair of duplicates greater than 2 is 2^20 = 1048576 = a(154) = a(231), located in antidiagonals 17 = A233328(2) and 21, respectively. For additional duplicate numbers in this sequence see A335903.  - _Hartmut F. W. Hoft_, Jun 29 2020

Alois P. Heinz, Antidiagonals n = 0..200, flattened (first 20 antidiagonals from Alec Jones)
Alec Jones, Proof that columns have finitely many odd entries.
Peter Kagey, Bitmap showing parity of first 1024 rows and 512 columns. (Odd values are white; even values are black.)
Peter Kagey, Animated example illustrating the first fifteen terms.

Cf. A064642 is analogous if a cell can only "see" its immediate neighbors.
Cf. A035002, A059450, A132439, A279966, A279967, A279211.
See A280026, A280027 for similar sequences based on a spiral.
Cf. A011782, A335903.

s[0, 0] = 1; s[i_, j_] := s[i, j] = Sum[s[k, j], {k, 0, i-1}] + Sum[s[i, k], {k, 0, j-1}] + Sum[s[i+j-k, k], {k, 0, j-1}] + Sum[s[i-k-1, j-k-1], {k, 0, Min[i, j] - 1}]
aDiag[m_] := Map[s[m-#, #]&, Range[0, m]]
a279212[n_] := Flatten[Map[aDiag, Range[0, n]]]
a279212[9] (* data - 10 antidiagonals;  Hartmut F. W. Hoft, Jun 29 2020 *)

T(0, 0) = 1; T(i, j) = Sum_{k=0..i-1} T(k, j) + Sum_{k=0..j-1} T(i, k) + Sum_{k=0..j-1} T(i+j-k, k) + Sum_{k=0..min(i, j)-1} T(i-k-1, j-k-1), with recursion upwards along antidiagonals. - Hartmut F. W. Hoft, Jun 29 2020

A132595 Number of ways to move a chess queen from the lower left corner to square (n,n), with the queen moving only up, right, or diagonally up-right.

Original entry on oeis.org

1, 3, 22, 188, 1712, 16098, 154352, 1499858, 14717692, 145509218, 1447187732, 14462966928, 145120265472, 1461040916988, 14751839744412, 149316973768398, 1514654852648052, 15393833895932658, 156716528008129892, 1597861126366223768

Offset: 1

Martin J. Erickson (erickson(AT)truman.edu), Nov 14 2007, Jan 28 2009

Main diagonal of the square array given in A132439.

a(n) is the number of Wythoff's Nim games starting with two equal piles of n stones. - Martin J. Erickson (erickson(AT)truman.edu), Dec 05 2008

			a(2) = 3 since the paths from (1,1) to (2,2) are
(1,1)->(2,1)->(2,2),
(1,1)->(1,2)->(2,2),
(1,1)->(2,2).

Alois P. Heinz, Table of n, a(n) for n = 1..300
M. Erickson, S. Fernando, K. Tran, Enumerating Rook and Queen Paths , Bulletin of the Institute for Combinatorics and Its Applications, Volume 60 (2010), 37-48.

Cf. A132439.

Column k=2 of A229345.

Rest[CoefficientList[Series[(x (x-1)/(3x-2))(1+(1-x)/Sqrt[1-12x+16x^2]),{x,0,20}],x]] (* Harvey P. Dale, Feb 09 2015 *)

G.f.: (x*(x-1)/(3*x-2))*(1 + (1-x)/sqrt(1 - 12*x + 16*x^2)). a(n) is asymptotic to (5^(3/4)*(sqrt(5)-2)/16)*(6+2*sqrt(5))^n/sqrt(Pi*n).

a(1)=1; a(2)=3; a(3)=22; a(4)=188; a(n) = ((29*n-47)*a(n-1) + (-95*n + 238)*a(n-2) + (116*n - 418)*a(n-3) + (-48*n + 240)*a(n-4))/(2*n-2), n >= 5. - Martin J. Erickson (erickson(AT)truman.edu), Nov 20 2007

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.

Original entry on oeis.org

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

Peter Kagey, Apr 12 2020

			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).

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.)

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.

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).

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.

Original entry on oeis.org

1, 1, 2, 2, 5, 10, 4, 13, 33, 63, 8, 32, 98, 240, 454, 16, 76, 269, 777, 1871, 3539, 32, 176, 702, 2295, 6420, 15314, 29008, 64, 400, 1768, 6393, 19970, 54758, 129825, 246255, 128, 896, 4336, 17088, 58342, 176971, 478662, 1129967, 2145722, 256, 1984, 10416

Offset: 1

Peter Kagey, Apr 12 2020

First row is A175962.

			Table begins:
n\k|  1   2     3      4       5        6         7          8
---+----------------------------------------------------------
  1|  1   2    10     63     454     3539     29008     246255
  2|  1   5    33    240    1871    15314    129825    1129967
  3|  2  13    98    777    6420    54758    478662    4266102
  4|  4  32   269   2295   19970   176971   1593093   14532881
  5|  8  76   702   6393   58342   536080   4965056   46345046
  6| 16 176  1768  17088  163041  1550809  14765863  140982374
  7| 32 400  4336  44280  440602  4332221  42373370  413689403
  8| 64 896 10416 111984 1159580 11771312 118190333 1179448443
For example, the T(2,2) = 5 sequences of permissible queen's moves from (1,1) to (2,2) are:
(1,1) -> (1,2) -> (2,2),
(1,1) -> (2,1) -> (1,2) -> (2,2),
(1,1) -> (2,1) -> (2,2),
(1,1) -> (2,1) -> (3,1) -> (2,2), and
(1,1) -> (3,1) -> (2,2).

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals)
Peter Kagey, Parity bitmap for first 1024 rows and columns. (Even and odd entries and represented by black and white pixels respectively.)

Cf. A175962.
Cf. A035002 (up, right), A059450 (right, up-left), A132439 (up, right, up-right), A279212 (up, right, up-left), A334016 (right, up-right, up-left).
A033877 is the analog for king moves. For both king and queen moves, A094727 is the length of the longest sequence of moves.

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A279212 Fill an array by antidiagonals upwards; in the top left cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that can be seen from that cell.

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A132595 Number of ways to move a chess queen from the lower left corner to square (n,n), with the queen moving only up, right, or diagonally up-right.

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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.

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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.

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