A278212 Maximum total number of possible moves that any number of bishops of the same color can make on an n X n chessboard.

0, 2, 8, 20, 38, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100, 8464, 8836, 9216

Offset

1,2

Comments

a(n) <= A275815(n).

Conjecture: a(n) = floor(A275815(n)/2). - Peter Kagey, Nov 24 2016

Links

Table of n, a(n) for n=1..50.

Michael S. Branicky, Example for n = 6.

Peter Kagey, Examples for 1 <= n <= 5.

Formula

Lim_{n->oo} a(n)/n^2 = 4. Putting bishops on the 4n-4 border locations shows that a(n) >= 4(n-2)^2. On the other hand, a(n) <= 4n^2 since each location is in the path of at most 4 bishops. - Chai Wah Wu, Nov 20 2016

a(n) <= 4*(n-2)^2 + 4 since the maximum number of possible moves on a single diagonal of length k >= 3 is 2*(k-2), which is achieved by placing bishops at either end of the diagonal. On a diagonal of length 2 only 1 move is possible with just one bishop. - Andrew Howroyd, Oct 31 2023

Example

The following 5 X 5 chessboard illustrates a(5) = 38:
   +---+---+---+---+---+
  5| B | B | B |   | B |
   +---+---+---+---+---+
  4|   |   |   |   | B |
   +---+---+---+---+---+
  3| B | B |   |   | B |
   +---+---+---+---+---+
  2|   |   |   |   | B |
   +---+---+---+---+---+
  1| B | B | B |   | B |
   +---+---+---+---+---+
     A   B   C   D   E
The bishops at A1, A5, B1, B5, E1, E2, E4, and E5 each have three moves; the bishops at A3, C1, C5, and E3 each have two moves; and the bishop at B3 has six moves.

Prog

(Python) # see link in A275815
(Python) # (using PuLP library) see links section of A278214. - Christian Sievers, Oct 31 2023

Crossrefs

Cf. A278211, A275815, A278213.

Sequence in context: A038460 A077588 A025219 * A032767 A294869 A305129

Adjacent sequences: A278209 A278210 A278211 * A278213 A278214 A278215

Keyword

nonn

Author

Peter Kagey, Nov 15 2016

Extensions

a(6) from Michael S. Branicky, Feb 12 2021

a(7)-a(50) from Christian Sievers, Oct 31 2023

Status

approved