A006566 Dodecahedral numbers: a(n) = n*(3*n - 1)*(3*n - 2)/2. (Formerly M5089)

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

Offset

0,3

Comments

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). - 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)

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Victor Meally, Letter to N. J. A. Sloane, no date.

Ed Pegg Jr, Dodecahedral 2024.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

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

a(n) = A006564(n) + A035006(n). - Peter M. Chema, May 04 2016

E.g.f.: x*(2 + 18*x + 9*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 04 2016

From Amiram Eldar, Jan 09 2024: (Start)

Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi - 3*log(3))/2 (A295421).

Sum_{n>=1} (-1)^(n+1)/a(n) = (12*log(2) - sqrt(3)*Pi)/3. (End)

Maple

A006566:=(1+16*z+10*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
CoefficientList[Series[x (1 + 16 x + 10 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2015 *)

Prog

(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
(Magma) [n*(3*n-1)*(3*n-2)/2: n in [0..40]]; // Vincenzo Librandi, Dec 11 2015

Crossrefs

Cf. A027907, A035006, A060544, A093485, A124388, A228887, A228888, A295421.

Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006564 (icosahedral numbers).

Sequence in context: A044207 A044588 A172221 * A205312 A268888 A211158

Adjacent sequences: A006563 A006564 A006565 * A006567 A006568 A006569

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley, Nov 23 2001

Status

approved