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A144945
Number of ways to place 2 queens on an n X n chessboard so that they attack each other.
2
0, 6, 28, 76, 160, 290, 476, 728, 1056, 1470, 1980, 2596, 3328, 4186, 5180, 6320, 7616, 9078, 10716, 12540, 14560, 16786, 19228, 21896, 24800, 27950, 31356, 35028, 38976, 43210, 47740, 52576, 57728, 63206, 69020, 75180, 81696, 88578, 95836, 103480, 111520
OFFSET
1,2
COMMENTS
a(n) gives the number of edges on a graph with n X n nodes where each node corresponds to a square on an n X n chessboard and there is an edge between two nodes if two queens placed on the corresponding squares attack each other.
In other words, number of edges in the n X n queen graph. - Eric W. Weisstein, Jun 19 2017
Number of ways to place two queens on the same row or column = A006002: b(n) = n*C(n,2) = n^2*(n-1)/2; number of ways to place two queens on the same diagonal (either SW-NE or NE-SW) = A000330 shifted by one: c(n) = n(n-1)*(2*n-1)/6; total: a(n) = 2*b(n)+2*c(n) = n*(5*n-1)*(n-1)/3.
Starting with "6" = binomial transform of [6, 22, 26, 10, 0, 0, 0, ...]. - Gary W. Adamson, Aug 12 2009
Also the Harary index of the n X n king graph. - Eric W. Weisstein, Jun 20 2017
LINKS
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Harary Index
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Queen Graph
FORMULA
a(n) = (n-1)*n*(5*n-1)/3.
From Bruno Berselli, Sep 27 2011: (Start)
G.f.: 2*x^2*(3+2*x)/(1-x)^4.
a(-n) = -A174814(n).
a(n) = a(n-1) + 2*A005475(n-1).
Sum_{i=1..n} a(i) = (n-1)*n*(n+1)*(5*n+2)/12. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4; a(1)=0, a(2)=6, a(3)=28, a(4)=76. - Harvey P. Dale, Oct 15 2011
a(n) = Sum_{i=1..n-1} i*(5*i+1), with a(0)=0, a(1)=6. - Bruno Berselli, Feb 10 2014
E.g.f.: x^2*(9+5*x)*exp(x)/3. - Robert Israel, Nov 02 2014
EXAMPLE
Example: For n=2 there are two rows, two columns and two diagonals. Each of these can be filled with two queens, giving a(2)=6.
For n=3 there are C(3,2) = 3 ways to place two queens on the same rows or column, giving C(3,2)*3 = 9 ways to place two queens on the same rows and 9 ways to place two queens on the same column. There are three nontrivial SW-NE diagonals, two of length two (each giving 1 way to place two attacking queens) and one of length three (giving 3 ways to place two attacking queens): total 3+1+1=5. There are also 5 ways to place two queens on the same NW-SE diagonal, giving a total of 9+9+5+5 = 28.
MAPLE
A144945:=n->(n-1)*n*(5*n-1)/3: seq(A144945(n), n=1..50); # Wesley Ivan Hurt, Nov 02 2014
MATHEMATICA
Table[n (5 n - 1) (n - 1)/3, {n, 50}] (* Harvey P. Dale, Oct 15 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 28, 76}, 50] (* Harvey P. Dale, Oct 15 2011 *)
CoefficientList[Series[2 x (3 + 2 x)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 07 2017 *)
PROG
(Magma) [(n-1)*n*(5*n-1)/3: n in [1..40]]; // Vincenzo Librandi, Sep 28 2011
(PARI) a(n) = (n-1)*n*(5*n-1)/3 \\ Charles R Greathouse IV, Jun 19 2017
(PARI) first(n) = Vec(2*x^2*(3+2*x)/(1-x)^4 + O(x^(n+1)), -n) \\ Iain Fox, Dec 07 2017
CROSSREFS
Sequence in context: A326134 A343512 A326484 * A308585 A202956 A279915
KEYWORD
nonn,easy,nice
AUTHOR
Paolo Bonzini, Sep 26 2008
EXTENSIONS
More terms from Harvey P. Dale, Oct 15 2011
STATUS
approved