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.

Previous Showing 21-23 of 23 results.

A195661 Number of ways to place 12n nonattacking kings on a vertical cylinder 24 X 2n.

Original entry on oeis.org

8192, 1270246, 44653028, 720390254, 7177627944, 51526819510, 291859775552, 1382652697282, 5700499630916, 21042965606234, 71028444904044, 222770819826574, 657397551407816, 1843639061043694, 4953451546255928, 12835026767559890, 32249277650536068
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2011

Keywords

Comments

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 24 are in contact (number of columns = 24, number of rows = 2n).

Links

Crossrefs

Cf. A195656, A195651, A137432.

Formula

Recurrence: a(n) = 4*a(n-13) - 48*a(n-12) + 265*a(n-11) - 891*a(n-10) + 2035*a(n-9) - 3333*a(n-8) + 4026*a(n-7) - 3630*a(n-6) + 2442*a(n-5) - 1210*a(n-4) + 429*a(n-3) - 103*a(n-2) + 15*a(n-1).
G.f.: -(1 + 8177*x + 1147469*x^2 + 26442685*x^3 + 177917014*x^4 + 436010362*x^5 + 423443926*x^6 + 163698250*x^7 + 23613841*x^8 + 1078869*x^9 + 9965*x^10 + 41*x^11)/((x-1)^11*(2*x-1)^2).
a(n) = (74405871551*n - 1097352668753)*2^n + 696317/2016*n^10 + 420699809/30240*n^9 + 66463031/210*n^8 + 26602370087/5040*n^7 + 33515235289/480*n^6 + 1076425504013/1440*n^5 + 32380230257101/5040*n^4 + 325331895133417/7560*n^3 + 29685456992323/140*n^2 + 72053208873316/105*n + 1097352668754.

A319096 Number of nonequivalent ways to place n^2 nonattacking kings on a 2n X 2n chessboard under all symmetry operations of the square.

Original entry on oeis.org

1, 14, 459, 35312, 4072108, 638653285, 128441726634, 31872148398195, 9490641145219266, 3321018871480028710
Offset: 1

Views

Author

Anton Nikonov, Dec 21 2018

Keywords

Comments

A maximum of n^2 nonattacking kings may be placed on a 2n X 2n chessboard.

Examples

			For n = 2 there are a(2) = 14 distinct solutions from 79 that will not be repeated at all possible turns and reflections.
------------
1.                  2.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
| * |   | * |   |   | * |   |   | * |
|   |   |   |   |   |   |   |   |   |
------------
3.                  4.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
| * |   |   |   |   |   | * |   | * |
|   |   |   | * |   |   |   |   |   |
------------
5.                  6.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
|   | * |   |   |   |   |   | * |   |
|   |   |   | * |   | * |   |   |   |
------------
7.                  8.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
|   |   |   | * |   |   |   |   |   |
| * |   |   |   |   | * |   | * |   |
------------
9.                  10.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   | * |
| * |   |   | * |   |   | * |   |   |
------------
11.                 12.
_________________   _________________
| * |   | * |   |   | * |   |   | * |
|   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   | * |   |   |
|   | * |   | * |   |   |   |   | * |
------------
13.                 14.
_________________   _________________
| * |   |   | * |   |   | * |   |   |
|   |   |   |   |   |   |   |   | * |
|   |   |   |   |   | * |   |   |   |
| * |   |   | * |   |   |   | * |   |
------------

Crossrefs

Cf. A018807 (rotations and reflections considered distinct).
Cf. A137432 (on cylindrical chessboard).
Cf. A236679, A322284, A321614.

Formula

a(n) = A236679(2n+1, n^2).

Extensions

a(4)-a(10) from Andrew Howroyd, Dec 21 2018

A195657 Number of ways to place 8n nonattacking kings on a vertical cylinder 16 X 2n.

Original entry on oeis.org

512, 17536, 218052, 1599820, 8500668, 36383284, 133538996, 437500380, 1314748124, 3694894500, 9849731140, 25173962492, 62193359676, 149475988116, 351246183572, 810197361564, 1840289301660, 4126688132548, 9154339355684, 20122502355004, 43888598831484
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2011

Keywords

Comments

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 16 are in contact (number of columns = 16, number of rows = 2n).

Links

Crossrefs

Cf. A195652, A174558, A137432.

Formula

Recurrence: a(n) = 4*a(n-9) - 32*a(n-8) + 113*a(n-7) - 231*a(n-6) + 301*a(n-5) - 259*a(n-4) + 147*a(n-3) - 53*a(n-2) + 11*a(n-1).
G.f.: -(1 + 501*x + 11957*x^2 + 52145*x^3 + 55651*x^4 + 13919*x^5 + 695*x^6 + 27*x^7)/((x-1)^7*(2*x-1)^2).
a(n) = (1751437*n - 15876635)*2^n + 8431/45*n^6 + 22263/5*n^5 + 500633/9*n^4 + 1381699/3*n^3 + 117001024/45*n^2 + 138801256/15*n + 15876636.
Previous Showing 21-23 of 23 results.

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

Content is available under The OEIS End-User License Agreement.