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A085250
4 times hexagonal numbers: a(n) = 4*n*(2*n-1).
23
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
0,2
COMMENTS
a(n) also can represented as n concentric squares (see example). - Omar E. Pol, Aug 21 2011
Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011
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
EXAMPLE
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)
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 *)
PROG
(PARI) a(n)=4*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jun 23 2003
EXTENSIONS
Edited by Don Reble, Nov 13 2005
Added zero, better definition, corrected offset and edited original formula. - Omar E. Pol, Dec 11 2008
STATUS
approved