%I M5089 #126 Jan 09 2024 08:47:29

%S 0,1,20,84,220,455,816,1330,2024,2925,4060,5456,7140,9139,11480,14190,

%T 17296,20825,24804,29260,34220,39711,45760,52394,59640,67525,76076,

%U 85320,95284,105995,117480,129766,142880,156849,171700,187460,204156

%N Dodecahedral numbers: a(n) = n*(3*n - 1)*(3*n - 2)/2.

%C Schlaefli symbol for this polyhedron: {5,3}.

%C A093485 = first differences; A124388 = second differences; third differences = 27. - _Reinhard Zumkeller_, Oct 30 2006

%C One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - _Daniel Forgues_, May 14 2010

%C From _Peter Bala_, Sep 09 2013: (Start)

%C 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:

%C Sum_{n >= 1} binomial(3*n,3)*x^n = (x + 16*x^2 + 10*x^3)/(1-x)^4;

%C Sum_{n >= 1} binomial(3*n+1,3)*x^n = (4*x + 19*x^2 + 4*x^3)/(1-x)^4;

%C Sum_{n >= 1} binomial(3*n+2,3)*x^n = (10*x + 16*x^2 + x^3)/(1-x)^4. (End)

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

%H T. D. Noe, <a href="/A006566/b006566.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.

%H Victor Meally, <a href="/A006556/a006556.pdf">Letter to N. J. A. Sloane</a>, no date.

%H Ed Pegg Jr, <a href="/A006566/a006566.jpg">Dodecahedral 2024</a>.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x(1 + 16x + 10x^2)/(1 - x)^4.

%F a(n) = A000292(3n-3) = A054776(n)/6 = n*A060544(n).

%F a(n) = C(n+2,3) + 16 C(n+1,3) + 10 C(n,3).

%F 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

%F 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

%F a(n) = A006564(n) + A035006(n). - _Peter M. Chema_, May 04 2016

%F E.g.f.: x*(2 + 18*x + 9*x^2)*exp(x)/2. - _Ilya Gutkovskiy_, May 04 2016

%F From _Amiram Eldar_, Jan 09 2024: (Start)

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

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

%p A006566:=(1+16*z+10*z**2)/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t Table[n(3n-1)(3n-2)/2,{n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 13 2011 *)

%t LinearRecurrence[{4,-6,4,-1},{0,1,20,84},40] (* _Harvey P. Dale_, Jul 24 2013 *)

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

%o (PARI) a(n)=n*(3*n-1)*(3*n-2)/2

%o (Haskell)

%o a006566 n = n * (3 * n - 1) * (3 * n - 2) `div` 2

%o a006566_list = scanl (+) 0 a093485_list -- _Reinhard Zumkeller_, Jun 16 2013

%o (Magma) [n*(3*n-1)*(3*n-2)/2: n in [0..40]]; // _Vincenzo Librandi_, Dec 11 2015

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

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

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Henry Bottomley_, Nov 23 2001