OFFSET
3,2
COMMENTS
Let I = I_n be the n X n identity matrix and P = P_n be the incidence matrix of the cycle (1,2,3,...,n).
Let T = P^(-1)+I+P.
11000...01
11100....0
01110.....
00111.....
..........
00.....111
10.....011
Then a(n) is the number of (0,1) n X n matrices A <= T (i.e., an element of A can be 1 only if T has a 1 at this place) having exactly two 1's in every row and column with per(A) = 4.
a(n) is the maximum number m such that m white kings and m black kings can coexist on an n+1 X n+1 chessboard without attacking each other. - Aaron Khan, Jul 05 2022
REFERENCES
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3 (1992), 15-19.
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..1000
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = (n - t(n))*(n - 3 + t(n))/2, where t(n) = 1-(n mod 2).
G.f.: x^4*(3-x)/( (1+x)*(1-x)^3 ). - R. J. Mathar, Mar 06 2011
From Bruno Berselli, Sep 13 2011: (Start)
a(n) + a(n+1) = A005563(n-2).
a(-n) = A084265(n). (End)
a(n) = 1 -2*n +floor(n/2) +floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
From Wesley Ivan Hurt, May 25 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
a(n) = Sum_{i=(-1)^n..n-2} i. (End)
With offset 0, this is ceiling(n/2)*(2*floor(n/2)+3). - N. J. A. Sloane, Jan 16 2020
E.g.f.: (1/2)*((1-x)*exp(x/2) - exp(-x/2))^2. - G. C. Greubel, Mar 22 2022
EXAMPLE
For n=5 the reference matrix is:
11001
11100
01110
00111
10011
There are 2^(3*n) = 32768 0-1 matrices obtained from removing one or more 1's in it.
There are 305 such matrices with permanent 4 and there are 13 such matrices with exactly two 1's in every column and every row.
There are 5 matrices having both properties. One of them is:
10001
01100
01100
00011
10010
From Aaron Khan, Jul 05 2022: (Start)
Examples of the sequence when used for kings on a chessboard:
.
A solution illustrating a(2)=3:
+-------+
| B B B |
| . . . |
| W W W |
+-------+
.
A solution illustrating a(3)=5:
+---------+
| B B B B |
| B . . . |
| . . . W |
| W W W W |
+---------+
(End)
MAPLE
MATHEMATICA
Table[(n^2 - 3*n + 1 + (-1)^n)/2, {n, 3, 100}] (* or *) CoefficientList[Series[x (x - 3)/((1 + x)*(x - 1)^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, May 25 2015 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 3, 5, 10}, 90] (* Harvey P. Dale, Jan 14 2024 *)
PROG
(Magma) [(n^2-3*n+1+(-1)^n)/2: n in [3..100]]; // Vincenzo Librandi, Mar 24 2011
(PARI) a(n)=(n^2-3*n+1+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 16 2015
(Sage) [n*(n-3)/2 + ((n+1)%2) for n in (3..60)] # G. C. Greubel, Mar 22 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Shevelev, Apr 12 2010
EXTENSIONS
Matrix class definition checked, edited and illustrated by Olivier Gérard, Mar 26 2011
STATUS
approved