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A007531
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n*(n-1)*(n-2) (or n!/(n-3)!).
(Formerly M4159)
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43
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0, 0, 0, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620, 50616, 54834, 59280, 63960, 68880
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OFFSET
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0,4
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COMMENTS
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Ed Pegg Jr conjectures that n^3 - n = k! has a solution iff n is 2, 3, 5 or 9 (when k is 3, 4, 5 and 6).
Three-dimensional promic (or oblong) numbers, cf. A002378 - Alexandre Wajnberg, Dec 29 2005
Doubled first differences of tritriangular numbers A050534(n) = (1/8)n(n + 1)(n - 1)(n - 2). a(n) = 2*(A050534(n+1) - A050534(n)). - Alexander Adamchuk, Apr 11 2006
If Y is a 4-subset of an n-set X then, for n>=6, a(n-4) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
Convolution of A005843 with A008585. [Reinhard Zumkeller, Mar 07 2009]
a(n) = A000578(n) - A000567(n). [Reinhard Zumkeller, Sep 18 2009]
For n>3: a(n) = A173333(n,n-3). [Reinhard Zumkeller, Feb 19 2010]
Let H be the n-by-n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row 2 of B equals (-1)^n a(n+1). - T. D. Noe, May 01 2011
a(n) equals 2^(n-1) times the coefficient of log(3) in 2F1(n-2,n-2,n,-2) [John M. Campbell, Jul 16 2011]
For n>2 a(n)=1/(Integral_{x=0..Pi/2} (sin(x))^5*(cos(x))^(2*n-5)). [Francesco Daddi, Aug 02 2011]
a(n) is the number of functions f:[3]->[n] that are injective since there are n choices for f(1), (n-1) choices for f(2), and (n-2) choices for f(3). Also, a(n+1) is the number of functions f:[3]->[n] that are width-2 restricted (that is, the pre-image under f of any element in [n] is of size 2 or less). See "Width-restricted finite functions" link below. [Dennis P. Walsh, Mar 01 2012]
This sequence is produced by three consecutive triangular numbers t(n-1), t(n-2) and t(n-3) in the expression 2*t(n-1)*(t(n-2)-t(n-3)) for n=0,1,2,... - J. M. Bergot, May 14, 2012
For n > 2: A020639(a(n)) = 2; A006530(a(n)) = A093074(n-1). - Reinhard Zumkeller, Jul 04 2012
Number of contact points between equal spheres arranged in a tetrahedron with n-1 spheres in each edge. [Ignacio Larrosa Cañestro, Jan 07 2013]
Also for n >= 3, area of Pythagorean triangle in which one side differs from hypotenuse by two units. Consider any Pythagorean triple (2n, n^2-1, n^2+1) where n > 1. The area of such a Pythagorean triangle is n(n^2-1). For n=2,3,4,.. the areas are 6,24,60,.... which are the given terms of the series. - Jayanta Basu, Apr 11 2013
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REFERENCES
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R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 40.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013
Dennis Walsh, Width-restricted finite functions
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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a(n) = 6*A000292(n-2).
Sum(Polygorial(3, i), i=1..n) - Daniel Dockery (peritus(AT)gmail.com), Jun 16, 2003
a(0) = a(1) = a(2) = 0, a(n) = 3a(n-1) - 3a(n-2) + a(n-3) + 6. - Zak Seidov, Feb 09 2006.
G.f.: 6*x^2/(1-x)^4.
a(-n) = -a(n+2).
1/6 + 3/24 + 5/60 +...= 3/4 [Jolley Eq. 213]
Other than the first two 0's this sequence is the same as the sequence a(n)=n^3-n. - Mohammad K. Azarian, Jul 26 2007
E.g.f.:x^3*exp(x). [Geoffrey Critzer, Feb 08 2009]
If the first 0 is eliminated, a(n)= floor(n^5/(n^2+1)) [Gary Detlefs, Feb 11 2010]
1/6+1/24+1/60+...+1/(n*(n+1)*(n+2))+...=1/4. [Mohammad K. Azarian, Dec. 29 2010]
a(0) = 0, a(n) = a(n-1) + 3*(n-1)*(n-2) - Jean-François Alcover, Jan 08 2013.
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MAPLE
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[seq(6*binomial(n, 3), n=0..41)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2006
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MATHEMATICA
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Table[n^3 - 3n^2 + 2n, {n, 0, 42}]
Table[FactorialPower[n, 3], {n, 0, 42}] (* Arkadiusz Wesolowski, Oct 29 2012 *)
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PROG
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(PARI) a(n)=n*(n-1)*(n-2)
(MAGMA) [n*(n-1)*(n-2): n in [0..40]]; // Vincenzo Librandi, May 02 2011
(Haskell)
a007531 n = product [n-2..n] - Reinhard Zumkeller, Jul 04 2012
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CROSSREFS
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Cf. A002378, A005563, A084939, A084940, A084941, A084942, A084943, A084944.
Cf. A007531.
Sequence in context: A086768 A160944 A160936 * A130669 A214308 A101854
Adjacent sequences: A007528 A007529 A007530 * A007532 A007533 A007534
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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STATUS
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approved
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