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A006566
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Dodecahedral numbers: n(3n-1)(3n-2)/2.
(Formerly M5089)
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22
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0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711, 45760, 52394, 59640, 67525, 76076, 85320, 95284, 105995, 117480, 129766, 142880, 156849, 171700, 187460, 204156
(list;
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OFFSET
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0,3
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COMMENTS
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Schlaefli symbol for this polyhedron: {5,3}
A093485 = first differences; A124388 = second differences; third differences = 27. - Reinhard Zumkeller, Oct 30 2006
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). [From Daniel Forgues, May 14 2010]
From Peter Bala, Sep 09 2013: (Start)
a(n) = binomial(3*n,3). Two related sequences are binomial(3*n+1,3) (A228887) and binomial(3*n+2,3) (A228888). The o.g.f.'s for these three sequences are rational functions whose numerator polynomials are obtained from the fourth row [1, 4, 10, 16, 19, 16, 10, 4, 1] of the triangle of trinomial coefficients A027907 by taking every third term:
sum(n >= 1, binomial(3*n,3)*x^n ) = (x + 16*x^2 + 10*x^3)/(1-x)^4
sum(n >= 1, binomial(3*n+1,3)*x^n ) = (4*x + 19*x^2 + 4*x^3)/(1-x)^4
sum(n >= 1, binomial(3*n+2,3)*x^n ) = (10*x + 16*x^2 + x^3)/(1-x)^4.
(End)
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REFERENCES
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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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T. D. Noe, Table of n, a(n) for n=0..1000
Tanya Khovanova, Recursive Sequences
Hyun Kwang Kim, On Regular Polytope Numbers
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).
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FORMULA
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G.f.: x(1+16x+10x^2)/(1-x)^4. a(n) = A000292(3n-3) = A054776(n)/6 = n*A060544(n).
a(n) = C(n+2,3) + 16 C(n+1,3) + 10 C(n,3)
a(0)=0, a(1)=1, a(2)=20, a(3)=84, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jul 24 2013
a(n) = binomial(3*n,3). a(-n) = - A228888(n). sum(n >= 1, 1/a(n) ) = 1/2*( sqrt(3)*Pi - 3*log(3) ). sum(n >= 1, (-1)^n/a(n) ) = 1/3*sqrt(3)*Pi - 4*log(2). - Peter Bala, Sep 09 2013
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MAPLE
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A006566:=(1+16*z+10*z**2)/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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Table[n(3n-1)(3n-2)/2, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 20, 84}, 40] (* Harvey P. Dale, Jul 24 2013 *)
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PROG
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(PARI) a(n)=n*(3*n-1)*(3*n-2)/2
(Haskell)
a006566 n = n * (3 * n - 1) * (3 * n - 2) `div` 2
a006566_list = scanl (+) 0 a093485_list -- Reinhard Zumkeller, Jun 16 2013
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CROSSREFS
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A027907, A228887, A228888.
Sequence in context: A044207 A044588 A172221 * A205312 A211158 A154077
Adjacent sequences: A006563 A006564 A006565 * A006567 A006568 A006569
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Henry Bottomley, Nov 23 2001
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STATUS
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approved
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