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A006564 Icosahedral numbers: a(n) = n*(5*n^2 -5*n + 2)/2.
(Formerly M4837)
12
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.

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(0)=1, a(1)=12, a(2)=48, a(3)=124, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - 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. A000292, A000566, A000578, A005900, A006566.

Sequence in context: A135453 A165280 A173548 * A239352 A265040 A059162

Adjacent sequences:  A006561 A006562 A006563 * A006565 A006566 A006567

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 3 18:47 EST 2016. Contains 278745 sequences.