%I M4837 #90 Sep 08 2022 08:44:35
%S 1,12,48,124,255,456,742,1128,1629,2260,3036,3972,5083,6384,7890,9616,
%T 11577,13788,16264,19020,22071,25432,29118,33144,37525,42276,47412,
%U 52948,58899,65280,72106,79392,87153,95404,104160,113436,123247,133608
%N Icosahedral numbers: a(n) = n*(5*n^2 - 5*n + 2)/2.
%C Schlaefli symbol for this polyhedron: {3,5}.
%C One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - _Daniel Forgues_, May 14 2010
%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="/A006564/b006564.txt">Table of n, a(n) for n = 1..1000</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., Vol. 131, No. 1 (2002), pp. 65-75.
%H Victor Meally, <a href="/A006556/a006556.pdf">Letter to N. J. A. Sloane</a>, no date.
%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = C(n+2,3) + 8*C(n+1,3) + 6*C(n,3).
%F 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
%F G.f.: x*(6*x^2 + 8*x + 1)/(x-1)^4. - _Harvey P. Dale_, May 26 2011
%F a(n) = A006566(n) - A035006(n). - _Peter M. Chema_, May 04 2016
%F E.g.f.: x*(2 + 10*x + 5*x^2)*exp(x)/2. - _Ilya Gutkovskiy_, May 04 2016
%F Sum_{n>=1} 1/a(n) = A175578. - _Amiram Eldar_, Jan 03 2022
%p A006564:=(1+8*z+6*z**2)/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t 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 *)
%o (Magma) [(5*n^3-5*n^2+2*n)/2: n in [1..100]] // _Vincenzo Librandi_, Nov 21 2010
%o (Haskell)
%o a006564 n = n * (5 * n * (n - 1) + 2) `div` 2
%o -- _Reinhard Zumkeller_, Jun 16 2013
%o (PARI) a(n)=5*n^2*(n-1)/2+n \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000566, A053012, A175578.
%Y Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers).
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_