A085250 4 times hexagonal numbers: a(n) = 4*n*(2*n-1).
0, 4, 24, 60, 112, 180, 264, 364, 480, 612, 760, 924, 1104, 1300, 1512, 1740, 1984, 2244, 2520, 2812, 3120, 3444, 3784, 4140, 4512, 4900, 5304, 5724, 6160, 6612, 7080, 7564, 8064, 8580, 9112, 9660, 10224, 10804, 11400, 12012, 12640, 13284
Offset: 0
Examples
From _Omar E. Pol_, Aug 21 2011: (Start) Illustration of initial terms as concentric squares: . . o o o o o o o o o o . o o . o o o o o o o o o o o o o o . o o o o o o . o o o o o o o o o o o o . o o o o o o o o o o o o . o o o o o o . o o o o o o o o o o o o o o . o o . o o o o o o o o o o . . 4 24 60 . (End)
Links
- Ivan Panchenko, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
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Mathematica
Table[8*n^2 - 4*n, {n, 0, 50}] (* G. C. Greubel, Jul 14 2017 *) 4 PolygonalNumber[6,Range[0,50]] (* Harvey P. Dale, Oct 19 2022 *) -
PARI
a(n)=4*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015
Formula
a(n) = A067239(n)/2, for n>0.
Sum_{n>0} 1/a(n) = log(2)/2.
a(n) = A000384(n)*4. - Omar E. Pol, Dec 11 2008
a(n) = 16*n + a(n-1) - 12 (with a(0)=0). - Vincenzo Librandi, Aug 08 2010
G.f.: 4*x*(1 + 3*x)/(1 - 3*x + 3*x^2 - x^3). - Colin Barker, Jan 04 2012
E.g.f.: 4*x*(2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A046092(2n-1), for n > 0. - Bruce J. Nicholson, Sep 04 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 - log(2)/4. - Amiram Eldar, Mar 17 2022
Extensions
Edited by Don Reble, Nov 13 2005
Added zero, better definition, corrected offset and edited original formula. - Omar E. Pol, Dec 11 2008
Comments