OFFSET
1,2
COMMENTS
Schlaefli symbol for this polyhedron: {3,5}.
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 113.
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 = 1..1000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Victor Meally, Letter to N. J. A. Sloane, no date. [but surely in the range 1973-1991]
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.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = C(n+2,3) + 8*C(n+1,3) + 6*C(n,3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=12, a(2)=48, a(3)=124. - Harvey P. Dale, May 26 2011
G.f.: x*(6*x^2 + 8*x + 1)/(x-1)^4. - Harvey P. Dale, May 26 2011
E.g.f.: x*(2 + 10*x + 5*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 04 2016
Sum_{n>=1} 1/a(n) = A175578. - Amiram Eldar, Jan 03 2022
MAPLE
A006564:=(1+8*z+6*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[n (5n^2-5n+2)/2, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 12, 48, 124}, 40] (* Harvey P. Dale, May 26 2011 *)
PROG
(Magma) [(5*n^3-5*n^2+2*n)/2: n in [1..100]]; // Vincenzo Librandi, Nov 21 2010
(Haskell)
a006564 n = n * (5 * n * (n - 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013
(PARI) a(n)=5*n^2*(n-1)/2+n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,nice,easy,changed
AUTHOR
STATUS
approved