Sangeet Paul - cp's OEIS frontend

A309442 Minimum number of colors needed to color the cells of the six regular convex polychora such that no two cells with a common face share the same color (in the order 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, 600-cell).

5, 4, 2, 3, 5, 3

Offset: 1

Sangeet Paul, Aug 03 2019

Here, cells are 3-dimensional polyhedra, and faces are 2-dimensional polygons.
The sequence is the 4-dimensional analog of A244951.
The sequence is also the minimum number of colors needed to color the vertices of the six regular convex polychora such that no two vertices with a common edge share the same color (in the order 5-cell, 16-cell, 8-cell, 24-cell, 600-cell, 120-cell).

			a(1) = 5, since in the 5-cell, each cell has a common face with every other cell (analogous to the tetrahedron, where each face has a common edge with every other face).
a(2) = 4, since in the 8-cell, each cell has a common face with every other cell except its "opposite" cell (analogous to the cube, where each face has a common edge with every other face except its opposite face).
a(3) = 2, since the 16-cell's dual graph has no odd-edge cycles (analogous to the octahedron's dual graph having no odd-edge cycles).
a(4) = 3, since the 24-cell has at least one 3-color solution, and its dual graph has a 3-vertex subgraph with no 2-color solution.
a(5) = 5, since the 120-cell has at least one 5-color solution, and its dual graph has a 30-vertex subgraph with no 4-color solution.
a(6) = 3, since the 600-cell has at least one 3-color solution, and its dual graph has a 5-vertex subgraph with no 2-color solution.

Cf. A244951, A273509.

A327132 Last cell visited by knight moves on a spirally numbered hexagonal board of edge-length n, moving to the lowest unvisited cell at each step.

Original entry on oeis.org

1, 1, 1, 34, 45, 76, 98, 135, 181, 234, 290, 338, 413, 487, 566, 654, 742, 823, 930, 1051, 1169, 1291, 1414, 1548, 1685, 1813, 1968, 2138, 2304, 2455, 2632, 2815, 3016, 3187, 3388, 3597, 3803, 4026, 4246, 4473, 4714, 4948, 5194, 5447, 5702, 5969, 6244, 6514

Offset: 1

Sangeet Paul, Aug 22 2019

nonn

A hexagonal board of edge-length 3, for example, is numbered spirally as:
.
      17--18--19
     /
    16   6---7---8
   /    /         \
  15   5   1---2   9
   \    \     /   /
    14   4---3  10
     \          /
      13--12--11
.
In Glinski's hexagonal chess, a knight (N) can move to these (o) cells:
.
      . . . . .
     . . o o . .
    . o . . . o .
   . o . . . . o .
  . . . . N . . . .
   . o . . . . o .
    . o . . . o .
     . . o o . .
      . . . . .
.
a(n) stays constant at 72085 for n >= 177 since 72085 is also the last cell visited by knight moves on a spirally numbered infinite hexagonal board, moving to the lowest unvisited cell at each step.

Sangeet Paul, Table of n, a(n) for n = 1..200
Chess variants, Glinski's Hexagonal Chess
Wikipedia, Hexagonal chess - Gliński's hexagonal chess

Cf. A308312, A327131.

A327131 Cells visited by knight moves on a spirally numbered infinite hexagonal board, moving to the lowest unvisited cell at each step.

Original entry on oeis.org

1, 20, 6, 9, 4, 8, 5, 10, 13, 2, 14, 7, 11, 22, 3, 15, 12, 23, 26, 29, 16, 19, 34, 54, 17, 31, 50, 47, 24, 21, 18, 32, 35, 55, 30, 27, 45, 68, 25, 42, 39, 36, 33, 53, 78, 48, 51, 76, 106, 49, 73, 28, 46, 43, 40, 37, 58, 84, 87, 60, 63, 41, 69, 72, 101, 67, 44

Offset: 1

Sangeet Paul, Aug 22 2019

The infinite hexagonal board is numbered spirally as:
.
      17--18--19...
     /
    16   6---7---8
   /    /         \
  15   5   1---2   9
   \    \     /   /
    14   4---3  10
     \          /
      13--12--11
.
In Glinski's hexagonal chess, a knight (N) can move to these (o) cells:
.
      . . . . .
     . . o o . .
    . o . . . o .
   . o . . . . o .
  . . . . N . . . .
   . o . . . . o .
    . o . . . o .
     . . o o . .
      . . . . .
.
This sequence is finite and ends at a(83966) = 72085 when the knight is "trapped".

Sangeet Paul, Table of n, a(n) for n = 1..83966
Scott R. Shannon, Image showing the 83965 steps of the knight's path. The green central dot is the starting square and the red dot, located on the edge at the 7:30 clock position, the final square. Blue dots show the twelve occupied squares surrounding the final square. The lowest unvisited square, 71301, is the yellow dot on the edge at the 9:00 clock position.
Chess variants, Glinski's Hexagonal Chess
Wikipedia, Hexagonal chess - Gliński's hexagonal chess

Cf. A316667, A327132.

A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

Original entry on oeis.org

1, 2, 7, 10, 19, 24, 37, 44, 61, 70, 91, 102, 127, 140, 169, 184, 217, 234, 271, 290, 331, 352, 397, 420, 469, 494, 547, 574, 631, 660, 721, 752, 817, 850, 919, 954, 1027, 1064, 1141, 1180, 1261, 1302, 1387, 1430, 1519, 1564, 1657, 1704, 1801, 1850, 1951, 2002

Offset: 1

Sangeet Paul, Aug 17 2019

			a(1) = 1
.
  o
.
a(2) = 2
.
   . .
  o . o
   . .
.
a(3) = 7
.
    o . o
   . . . .
  o . o . o
   . . . .
    o . o
.
a(4) = 10
.
     . . . .
    o . o . o
   . . . . . .
  o . o . o . o
   . . . . . .
    o . o . o
     . . . .
.

Chess variants, Glinski's Hexagonal Chess
Wikipedia, Hexagonal chess - Gliński's hexagonal chess
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A003215, A049450.

Partial sums of A133090.

nn:=51; CoefficientList[Series[- (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2),{x, 0, nn}], x] (* Georg Fischer, May 10 2020 *)

a(n) = n^2 - (n\2) - (n\2)^2; \\ Andrew Howroyd, Aug 17 2019

def A309805(n): return n**2-(m:=n>>1)*(m+1) # Chai Wah Wu, Apr 04 2024

a(n) = n^2 - floor(n/2) - floor(n/2)^2.
From Stefano Spezia, Aug 18 2019 (Start)
G.f.: - (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2).
E.g.f.: (1/8)*exp(-x)*(-1 + 2*x + exp(2*x)*(1 + 4*x + 6*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
a(n) = (1/16)*(3 + (-1)^(1+2*n) - 4*n + 12*n^2 - 2*(-1)^n*(1 + 2*n)).
a(2*n-1) = A003215(n).
a(2*n) = A049450(n).
(End)

A308385 a(n) is the last square visited by fers moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 3, 15, 29, 61, 87, 139, 177, 249, 299, 391, 453, 565, 639, 771, 857, 1009, 1107, 1279, 1389, 1581, 1703, 1915, 2049, 2281, 2427, 2679, 2837, 3109, 3279, 3571, 3753, 4065, 4259, 4591, 4797, 5149, 5367, 5739, 5969, 6361, 6603, 7015, 7269, 7701, 7967, 8419

Offset: 1

Sangeet Paul, May 23 2019

A 5 X 5 board, for example, is numbered with the square spiral:
.
  21--22--23--24--25
   |
  20   7---8---9--10
   |   |           |
  19   6   1---2  11
   |   |       |   |
  18   5---4---3  12
   |               |
  17--16--15--14--13
.
A fers is a (1,1)-leaper and can move one square diagonally.

Colin Barker, Table of n, a(n) for n = 1..1000
Stephen Emmerson and Geoff Foster, A glossary of fairy chess definitions, British Chess Problem Society, 2018.
Wikipedia, Ferz
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A054552, A054556, A316667, A316884.

[(3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2: n in [1..50]]; // Vincenzo Librandi, Aug 01 2019

Table[(3/2) (5 + (-1)^n) - (10 + (-1)^n) n + 4 n^2, {n, 60}] (* Vincenzo Librandi, Aug 01 2019 *)

Vec(x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, May 23 2019

a(n) = (4n^2-9n+6)*[n is odd] + (4n^2-11n+9)*[n is even] where [] is the Iverson bracket.
a(n) = A054556(n)*[n is odd] + (A054552(n)+1)*[n is even] where [] is the Iverson bracket.
a(n) = A316884(n^2)*[n is odd] + A316884(n^2-n)*[n is even] where [] is the Iverson bracket.
From Colin Barker, May 23 2019: (Start)
G.f.: x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = (3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)
E.g.f.: (1/2)*exp(-x)*(3 + 2*x + exp(2*x)*(15 - 12*x + 8*x^2)) - 9. - Stefano Spezia, Aug 17 2019

A309260 Number of ways of placing 2n-1 nonattacking rooks on a hexagonal board with edge-length n in Glinski's hexagonal chess, inequivalent up to rotations and reflections of the board.

Original entry on oeis.org

1, 1, 1, 5, 29, 224, 3012, 55200, 1259794, 35488536, 1200819600

Offset: 1

Sangeet Paul, Jul 19 2019

A rook in Glinski's hexagonal chess can move to any cell along the perpendicular bisector of any of the 6 edges of the hexagonal cell it's on (analogous to a rook in orthodox chess which can move to any cell along the perpendicular bisector of any of the 4 edges of the square cell it's on).

			a(1) = 1
.
  o
.
a(2) = 1
.
   o .
  . . o
   o .
.
a(3) = 1
.
    o . .
   . . o .
  . . . . o
   o . . .
    . o .
.
a(4) = 5
.
     o . . .        o . . .        o . . .        . o . .        . o . .
    . . o . .      . . o . .      . . . o .      o . . . .      . . . . o
   . . . . o .    . . . . . o    . . . . . o    . . . . . o    o . . . . .
  . . . . . . o  . o . . . . .  . . o . . . .  . . . o . . .  . . . o . . .
   o . . . . .    . . . . . o    o . . . . .    . . . . . o    . . . . . o
    . o . . .      . . o . .      . . . . o      o . . . .      o . . . .
     . . o .        o . . .        . o . .        . o . .        . . o .
.

Alain Brobecker, Non Attacking Rooks on Hexhex and Triangular boards
Chess variants, Glinski's Hexagonal Chess
Vaclav Kotesovec, All inequivalent solutions for n = 2,3,4 and 5
Wikipedia, Hexagonal chess - Gliński's hexagonal chess

Cf. A000903, A002047, A003215, A309746, A309669.

a(1)-a(7) confirmed by Vaclav Kotesovec, Aug 16 2019
a(8) from Alain Brobecker, Dec 10 2021
a(8) confirmed by Vaclav Kotesovec, Dec 12 2021
a(9) from Alain Brobecker, Dec 13 2021
a(9) confirmed by Vaclav Kotesovec, Dec 18 2021
a(10)-a(11) from Bert Dobbelaere, Oct 24 2022

A308312 a(n) is the last square visited by knight moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 1, 14, 30, 69, 108, 150, 205, 264, 333, 408, 475, 553, 659, 763, 881, 1004, 1134, 1274, 1418, 1641, 1811, 1986, 2167, 2358, 2557, 2633, 2978, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084, 2084

Offset: 1

Sangeet Paul, May 19 2019

nonn

A 5 X 5 board, for example, is numbered with the square spiral:
.
  21--22--23--24--25
   |
  20   7---8---9--10
   |   |           |
  19   6   1---2  11
   |   |       |   |
  18   5---4---3  12
   |               |
  17--16--15--14--13
.
a(n) stays constant at 2084 for (2n-1) >= 57 since 2084 is also the last square visited by knight moves on a spirally numbered doubly infinite board, moving to the lowest available unvisited square at each step.

Cf. A316667.

A307973 Numbers beginning with a vowel in Hindi.

Original entry on oeis.org

1, 8, 18, 19, 21, 28, 29, 31, 38, 39, 41, 48, 49, 51, 58, 59, 61, 68, 69, 71, 78, 79, 80, 81, 88, 91, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130

Offset: 1

Sangeet Paul, May 30 2019

For 99 < n < 1000, n belongs to the list if (floor(n/100) mod 10) belongs to the list (i.e., is either 1 or 8).

For 10^(2k+1)-1 < n < 10^(2k+3), where k is a positive integer, n belongs to the list if (floor(n/(10^(2k+1))) mod 100) belongs to the list.

Cf. A000852.

cp's OEIS Frontend

User: Sangeet Paul

A309442 Minimum number of colors needed to color the cells of the six regular convex polychora such that no two cells with a common face share the same color (in the order 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, 600-cell).

Views

Author

Keywords

Comments

Examples

Crossrefs

A327132 Last cell visited by knight moves on a spirally numbered hexagonal board of edge-length n, moving to the lowest unvisited cell at each step.

Views

Author

Keywords

Comments

Links

Crossrefs

A327131 Cells visited by knight moves on a spirally numbered infinite hexagonal board, moving to the lowest unvisited cell at each step.

Views

Author

Keywords

Comments

Links

Crossrefs

A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A308385 a(n) is the last square visited by fers moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A309260 Number of ways of placing 2n-1 nonattacking rooks on a hexagonal board with edge-length n in Glinski's hexagonal chess, inequivalent up to rotations and reflections of the board.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A308312 a(n) is the last square visited by knight moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

Views

Author

Keywords

Comments

Crossrefs

A307973 Numbers beginning with a vowel in Hindi.

Views

Author

Keywords

Comments

Links

Crossrefs