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A035287 Number of ways to place a non-attacking white and black rook on n X n chessboard. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 (mn)^2 of the term (1/m^2 - 1/n^2) = (2n-1)/(mn)^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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site. [From Johannes W. Meijer, Jul 21 2009]

FORMULA

a(n) = n^2 * (n-1)^2.

a(n) = ( A002378(n-1) )^2. - Zerinvary Lajos, Apr 11 2006

From Stephen Crowley, Jul 19 2009: (Start)

a(n) = limit(n!*(2*n+1)/((d^n/dx^n)(1 + polylog(2,x) - polylog(2,x)/x), x=0);

Integral_{x=0..1} (1 + polylog(2,x) - polylog(2,x)/x) dx = Zeta(2) - Zeta(3).

Sum_{n >= 2} 1/a(n) = 2*Zeta(2) - 3 = A145426. (End)

a(n) = 4*A000537(n-1) = 2*A163102(n-1). - Omar E. Pol, Nov 29 2011

G.f.: 4*x^2*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 04 2012

a(n) = 4*( A000217(n-1) )^2. - J. M. Bergot, Nov 01 2012

E.g.f.: x^2*(2 + 4*x + x^2)*exp(x). - Ilya Gutkovskiy, May 24 2016

MATHEMATICA

Table[(n - 1)^2 n^2, {n, 30}] (* Alonso del Arte, May 20 2011 *)

PROG

(Sage) [n^2*(n-1)^2 for n in xrange(1, 35)] # Zerinvary Lajos, Dec 03 2009

(MAGMA) [n^2 * (n-1)^2: n in [1..40]]; // Vincenzo Librandi, May 21 2011

CROSSREFS

Cf. A002378.

Sequence in context: A296948 A125756 A173961 * A183354 A204504 A083223

Adjacent sequences:  A035284 A035285 A035286 * A035288 A035289 A035290

KEYWORD

nonn,easy

AUTHOR

Erich Friedman

STATUS

approved

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Last modified November 20 06:06 EST 2018. Contains 317385 sequences. (Running on oeis4.)