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 A006564 Icosahedral numbers: a(n) = n*(5*n^2 - 5*n + 2)/2. (Formerly M4837) 20
 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 T. D. Noe, Table of n, a(n) for n = 1..1000 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. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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 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 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. A000566. Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers). Sequence in context: A165280 A280058 A173548 * A239352 A292022 A265040 Adjacent sequences:  A006561 A006562 A006563 * A006565 A006566 A006567 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 17 23:33 EDT 2019. Contains 328135 sequences. (Running on oeis4.)