OFFSET
1,2
COMMENTS
The sequence 1, 4, 5, 8, 13, ... with g.f. -x*(1 +2*x -2*x^2 +2*x^3 +x^4)/ ((1+x) *(x^2+1) *(x-1)^3) and a(n)= 2*a(n-1) -a(n-2) +a(n-4) -2*a(n-5) +a(n-6) is a lower bound for a(n) achieved by packing 2x2 squares with 1's and 2x2 squares with 0's in a checkerboard pattern into the chessboard. - R. J. Mathar, Dec 03 2022
FORMULA
Conjectures:
a(n) = n^2/2 for n == 0 (mod 4).
a(n) = (n^2 + 1)/2 for n == 1 or 3 (mod 4).
a(n) = n^2/2 + 2 for n == 2 (mod 4).
EXAMPLE
For n = 1, a(1) = (1^2 + 1)/2 = 1
1
For n = 2, a(2) = (2^2)/2 + 2 = 4
11
11
For n = 3, a(3) = (3^2 + 1)/2 = 5
Starting here the solutions are not unique. We can mix 2X2 blocks from and S shapes along the diagonals.
110
110
001
or
110
011
001
For n = 4, a(4) = (4^2)/2 = 8
1100
1100
0011
0011
or
1100
0110
0011
1001
For n = 5, a(5) = (5^2 + 1)/2 = 13
11001
11001
00110
00110
11001
or
11001
01100
00110
10011
11001
For n = 6, a(6) = (6^6)/2 + 2 = 20
110011
110011
001100
001100
110011
110011
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Yang Hong, Oct 20 2021
STATUS
approved