OFFSET

1,2

COMMENTS

After the first 7 terms, the first differences are terms of A052928: for n >= 8, a(n) - a(n-1) = A052928(n-1).

The increase in differences going from an even n to an odd n, but not from an odd n to an even n, is due to the differing optimal layouts for odd vs. even n values. See example section for a(7) and a(8).

LINKS

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

For n > 6, a(n) = floor(((n-1)^2)/2).

G.f.: x^2*(2 - x + 2*x^3 - 2*x^4 - x^5 + 2*x^6 + 2*x^7 - 2*x^8)/((1 - x)^3*(1 + x)). - Stefano Spezia, Jul 05 2022

EXAMPLE

Examples for n=2 to n=6 have been included as they do not follow the general formula.

.

A solution illustrating a(2)=2:

+-----+

| B B |

| W W |

+-----+

.

A solution illustrating a(3)=3:

+-------+

| . . . |

| B B W |

| W W B |

+-------+

.

A solution illustrating a(4)=6:

+---------+

| B B . W |

| W W . B |

| B B . W |

| W W . B |

+---------+

.

A solution illustrating a(5)=10:

+-----------+

| W B W B W |

| W B W B W |

| . . . . . |

| B W B W B |

| B W B W B |

+-----------+

.

A solution illustrating a(6)=14:

+-------------+

| B B W W B B |

| W W B B W W |

| B . . . . B |

| W . . . . W |

| B B W W B B |

| W W B B W W |

+-------------+

.

Examples for n=7 and n=8 are provided, as while both follow the same formula, the layout for even values of n differs from the layout for odd values of n (related to the fact that, for even values of n, the floor function rounds down a non-integer value).

.

A solution illustrating a(7)=18:

+---------------+

| B B B B B B B |

| B B B B B B B |

| B . B . B . B |

| . . . . . . . |

| W . W . W . W |

| W W W W W W W |

| W W W W W W W |

+---------------+

.

A solution illustrating a(8)=24:

+-----------------+

| B B B B B B B B |

| B B B B B B B B |

| B B B B B B B B |

| . . . . . . . . |

| . . . . . . . . |

| W W W W W W W W |

| W W W W W W W W |

| W W W W W W W W |

+-----------------+

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Aaron Khan, Jul 04 2022

STATUS

approved