cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A287866 a(n) = n - A274933(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7
Offset: 1

Views

Author

N. J. A. Sloane, Jun 04 2017

Keywords

Comments

It would be nice to know how fast this grows.

Crossrefs

Cf. A274933, A287864, A287867.

Extensions

a(46)-a(99) from Andy Huchala, Mar 28 2024

A287864 Consider a symmetric pyramid-shaped chessboard with rows of squares of lengths n, n-2, n-4, ..., ending with either 2 or 1 squares; a(n) is the maximal number of mutually non-attacking queens that can be placed on this board.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23
Offset: 1

Views

Author

N. J. A. Sloane, Jun 03 2017

Keywords

Comments

Since there can be at most one queen per row, for n >= 2, a(n) <= floor(n/2). It would be nice to know how fast this sequence grows. Compare A287867.
If n=2t, the board contains t(t+1) squares; if n=2t+1 it contains (t+1)^2 squares. The number of squares is thus given by the quarter-squares sequence (A002620(n+1)).
The bisection a(2n+1) is A274933. - Rob Pratt, Jun 04 2017
For n = 1 to 100, here are the exceptions to the pattern that the values increase by 1 every two steps:
a(1) = a(2) = a(3) = 1
a(12) = a(13) = a(14) = 6
a(27) = a(28) = a(29) = 13
a(44) = a(45) = a(46) = 21
a(59) = a(60) = a(61) = 28
a(74) = a(75) = a(76) = 35
a(89) = a(90) = a(91) = 42. - Rob Pratt, Jun 04 2017

Examples

			Q = queen, X = empty square
---
Q   a(1)=1
---
QX   a(2)=1
---
.X.
QXX   a(3)=1
---
.QX.
XXXQ   a(4)=2
----
..X..
.QXX.
XXXQX   a(5)=2
-----
..QX..
.XXXQ.
XQXXXX   a(6)=3
------
...X...
..QXX..
.XXXQX.
XQXXXXX   a(7)=3
-------
...QX...
..XXXQ..
.XQXXXX.
XXXXQXXX   a(8)=4
--------
....QX....
...XXXQ...
..XQXXXX..
.XXXXQXXX.
XXQXXXXXXX   a(10)=5
----------
.....QX.....
....XXXQ....
...XQXXXX...
..XXXXQXXX..
.XXQXXXXXXX.
XXXXXXXXXQXX   a(12)=6
------------
......QX......
.....XXXQ.....
....XQXXXX....
...XXXXQXXX...
..XXQXXXXXXX..
.XXXXXXXXXQXX.
XXXXXXXXXXXXXX   a(14)=6
--------------

Links

Crossrefs

Cf. A002620, A274616, A274933, A287867.

Extensions

a(15)-a(100) from Rob Pratt, Jun 04 2017

A352241 Maximal number of nonattacking black-square queens on an n X n chessboard.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47
Offset: 1

Views

Author

George Baloglou, Mar 09 2022

Keywords

Links

Crossrefs

Cf. A030978, A053757, A190394, A191236, A274616, A274933.
Cf. this sequence (maximal number for black-squares), A352325 (black-squares counts), A352426 (maximal number for white-squares), A352599 (white-squares counts).

Formula

Conjecture: a(5k)=4k-1, a(5k+1)=4k, a(5k+2)=4k+1, a(5k+3)=4k+1, a(5k+4)=4k+2. [This does not hold for n = 52 and n = 57. - Andy Huchala, Apr 02 2024]
a(n) = A053757(n-1), at least for 1 <= n <= 12. [This is unlikely to continue. - N. J. A. Sloane, Mar 11 2022] [Indeed the equality does not hold for n=13. - Martin Ehrenstein, Mar 11 2022]
a(n+1) >= a(n); a(2n) = A352426(2n). - Martin Ehrenstein, Mar 23 2022

Extensions

a(13)-a(26) from Martin Ehrenstein, Mar 11 2022
a(27)-a(28) from Martin Ehrenstein, Mar 15 2022
a(29)-a(30) from Martin Ehrenstein, Mar 23 2022
a(31)-a(60) from Andy Huchala, Mar 27 2024

A287867 a(n) = floor(n/2) - A287864(n).

Original entry on oeis.org

-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
Offset: 1

Views

Author

N. J. A. Sloane, Jun 05 2017

Keywords

Comments

It would be nice to know how fast this sequence grows.

Links

Crossrefs

Cf. A287864, A274933, A287866.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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