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 A176222 a(n) = (n^2 - 3*n + 1 + (-1)^n)/2. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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) a(n) = A174239(n-2) * A174239(n-1). - Paul Curtz, Jul 17 2017 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 A176222:=n->(n^2-3*n+1+(-1)^n)/2: seq(A176222(n), n=3..100); # Wesley Ivan Hurt, May 25 2015 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 *) 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 Cf. A000211, A052928, A128209, A250000 (queens on a chessboard), A002620 (rooks on a chessboard), A355509 (knights on a chessboard). Sequence in context: A308805 A001841 A266793 * A008610 A281688 A078411 Adjacent sequences: A176219 A176220 A176221 * A176223 A176224 A176225 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

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Last modified December 5 19:04 EST 2022. Contains 358588 sequences. (Running on oeis4.)