

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
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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 fourline Latin rectangles, DAN Ukrainy, 3 (1992), 1519.


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*(3x)/( (1+x)*(1x)^3 ).  R. J. Mathar, Mar 06 2011
From Bruno Berselli, Sep 13 2011: (Start)
a(n) + a(n+1) = A005563(n2).
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(n1)  2*a(n3) + a(n4), n>4.
a(n) = Sum_{i=(1)^n..n2} i. (End)
a(n) = A174239(n2) * A174239(n1).  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)*((1x)*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 01 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^23*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^23*n+1+(1)^n)/2: n in [3..100]]; // Vincenzo Librandi, Mar 24 2011
(PARI) a(n)=(n^23*n+1+(1)^n)/2 \\ Charles R Greathouse IV, Oct 16 2015
(Sage) [n*(n3)/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



