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A035287
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Number of ways to place a non-attacking white and black rook on n X n chessboard.
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20
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0, 4, 36, 144, 400, 900, 1764, 3136, 5184, 8100, 12100, 17424, 24336, 33124, 44100, 57600, 73984, 93636, 116964, 144400, 176400, 213444, 256036, 304704, 360000, 422500, 492804, 571536, 659344, 756900, 864900, 984064, 1115136, 1258884
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OFFSET
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1,2
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COMMENTS
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a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for fixed different x_1, x_2 in {1,2,3,4} and fixed y_1, y_2 in {1,2,...,n} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 17 2007
The third differences of certain values of the hypergeometric function 3F2 lead to this sequence, i.e., 3F2([1,n+1,n+1], [n+2,n+2], z=1) - 3*3F2([1,n+2,n+2], [n+3,n+3], z=1) + 3*3F2([1,n+3,n+3], [n+4,n+4], z=1) - 3F2([1,n+4,n+4], [n+5,n+5], z=1) = (1/((n+2)*(n+3)))^2 with n = -1, 0, 1, 2, ... See also A162990. - Johannes W. Meijer, Jul 21 2009
a(n) is the denominator (m*n)^2 of the term (1/m^2 - 1/n^2) = (2*n-1)/(m*n)^2, n = m+1, m > 0 in the Rydberg formula, while A005408 is the numerator 2n-1. So the quotient A005408/A035287 simulates the hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013
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LINKS
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FORMULA
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a(n) = n^2 * (n-1)^2.
a(n) = n!*(2*n+1) / lim_{x->0} (d^n/dx^n) (polylog(2,x)*(1-1/x));
Sum_{n >= 2} 1/a(n) = 2*zeta(2) - 3 = A145426. (End) [Comment from Jianing Song, Dec 31 2022: Note that polylog(2,x)*(1-1/x) = -1 + Sum_{n>=1} ((2*n+1)/(n^2*(n+1)^2))*x^n, so (d^n/dx^n) (polylog(2,x)*(1-1/x)) = n!*(2*n+1)/(n^2*(n+1)^2) for n >= 1. - Jianing Song, Dec 31 2022]
G.f.: 4*x^2*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 04 2012
Sum_{n>=2} (-1)^n/a(n) = 3 - 4*log(2). - Amiram Eldar, Jul 02 2020
Product_{n>=2} (1 - 1/a(n)) = -cos(sqrt(5)*Pi/2)*cosh(sqrt(3)*Pi/2)/Pi^2. - Amiram Eldar, Jan 29 2021
(n^2)^2 + (n^2+1)^2 + ... + (n^2 + n)^2 + a(n) = (n^2 + n + 1)^2 + ... + (n^2 + 2*n)^2. - Charlie Marion, Jun 18 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n >= 6. - Jianing Song, Dec 30 2022
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MATHEMATICA
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PROG
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(Python)
for n in range(100):
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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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STATUS
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approved
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