A176222 a(n) = (n^2 - 3*n + 1 + (-1)^n)/2.

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

Offset: 3

Author

Vladimir Shevelev, Apr 12 2010

Keywords

• nonn
• easy

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

Examples

			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)
		

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).

Crossrefs

Cf. A000211, A052928, A128209, A250000 (queens on a chessboard), A002620 (rooks on a chessboard), A355509 (knights on a chessboard).

Programs

Magma

[(n^2-3*n+1+(-1)^n)/2: n in [3..100]]; // Vincenzo Librandi, Mar 24 2011

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 *)
LinearRecurrence[{2,0,-2,1},{0,3,5,10},90] (* Harvey P. Dale, Jan 14 2024 *)

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

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

Extensions

Matrix class definition checked, edited and illustrated by Olivier Gérard, Mar 26 2011