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

0,2

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

Links

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Crossrefs

Cf. A067239, A000384, A033581, A046092, A152734, A152751, A194274.

Sequence in context: A191778 A157625 A128205 * A166870 A124350 A112611

Adjacent sequences: A085247 A085248 A085249 * A085251 A085252 A085253

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