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A176222
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a(n) = (n^2 - 3*n + 1 + (-1)^n)/2.
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9
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0, 3, 5, 10, 14, 21, 27, 36, 44, 55, 65, 78, 90, 105, 119, 136, 152, 171, 189, 210, 230, 253, 275, 300, 324, 351, 377, 406, 434, 465, 495, 528, 560, 595, 629, 666, 702, 741, 779, 820, 860, 903, 945, 990, 1034, 1081, 1127, 1176, 1224, 1275, 1325, 1378, 1430
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OFFSET
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3,2
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COMMENTS
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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
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REFERENCES
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V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3 (1992), 15-19.
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LINKS
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FORMULA
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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
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
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EXAMPLE
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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
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)
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(Sage) [n*(n-3)/2 + ((n+1)%2) for n in (3..60)] # G. C. Greubel, Mar 22 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Matrix class definition checked, edited and illustrated by Olivier Gérard, Mar 26 2011
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STATUS
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approved
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