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A006564 Icosahedral numbers: a(n) = n*(5*n^2 - 5*n + 2)/2.
(Formerly M4837)
23
1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, 6384, 7890, 9616, 11577, 13788, 16264, 19020, 22071, 25432, 29118, 33144, 37525, 42276, 47412, 52948, 58899, 65280, 72106, 79392, 87153, 95404, 104160, 113436, 123247, 133608 (list; graph; refs; listen; history; text; internal format)
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
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
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.
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
a(n) = A006566(n) - A035006(n). - Peter M. Chema, May 04 2016
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
Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers).
Sequence in context: A165280 A280058 A173548 * A239352 A292022 A265040
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified March 19 03:30 EDT 2024. Contains 370952 sequences. (Running on oeis4.)