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A007531
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n*(n-1)*(n-2) (or n!/(n-3)!).
(Formerly M4159)
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39
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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 (alexandre.wajnberg(AT)skynet.be), 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 R. Janjic (agnus(AT)blic.net), Dec 28 2007
Convolution of A005843 with A008585. [From Reinhard Zumkeller, Mar 07 2009]
a(n) = A000578(n) - A000567(n). [From Reinhard Zumkeller, Sep 18 2009]
For n>3: a(n) = A173333(n,n-3). [From 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) [From 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)). [From Francesco Daddi, Aug 02 2011]
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REFERENCES
| 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
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| 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) [From Geoffrey Critzer, Feb 08 2009]
If the first 0 is eliminated, a(n)= floor(n^5/(n^2+1)) [From Gary Detlefs, Feb 11 2010]
1/6+1/24+1/60+...+1/(n*(n+1)*(n+2))+...=1/4. [From Mohammad K. Azarian, Dec. 29 2010]
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MAPLE
| [seq(6*binomial(n, 3), n=0..41)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2006
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MATHEMATICA
| Table[n^3 - 3n^2 + 2n, {n, 0, 41}]
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PROG
| (PARI) a(n)=n*(n-1)*(n-2)
(MAGMA) [n*(n-1)*(n-2): n in [0..40]]; // Vincenzo Librandi, May 02 2011
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CROSSREFS
| Cf. A002378, A005563, A084939, A084940, A084941, A084942, A084943, A084944.
Cf. A007531.
Sequence in context: A086768 A160944 A160936 * A130669 A101854 A101877
Adjacent sequences: A007528 A007529 A007530 * A007532 A007533 A007534
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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