A016743 Even cubes: a(n) = (2*n)^3.

0, 8, 64, 216, 512, 1000, 1728, 2744, 4096, 5832, 8000, 10648, 13824, 17576, 21952, 27000, 32768, 39304, 46656, 54872, 64000, 74088, 85184, 97336, 110592, 125000, 140608, 157464, 175616, 195112, 216000, 238328, 262144, 287496, 314432

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is also the number of non-degenerate triangles that can be drawn with vertices on a cross with n points on each branch. - James P. B. Hall, Nov 22 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Hilko Koning, 216 neodymium magnets for n=3.
• Ana Rechtman, Mars 2022, 1er défi, Images des Mathématiques, CNRS, 2022 (in French).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Even bisection of A000578, cf. A016755.

Cf. A016803 (even bisection), A016827 (odd bisection), A033581, A276712.

Programs

Magma

[(2*n)^3: n in [0..50]]; // Vincenzo Librandi, Sep 05 2011

Maple

A016743:=n->(2*n)^3: seq(A016743(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2018

Mathematica

Range[0, 78, 2]^3 (* Alonso del Arte, Apr 06 2013 *)

PARI

a(n) = 8*n^3; \\ Joerg Arndt, Apr 07 2013

Formula

a(n) = (2*n)^3 = 8*n^3.

G.f.: x*(8+32*x+8*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 02 2012

E.g.f.: 8*x*(1 +3*x +x^2)*exp(x). - G. C. Greubel, Sep 15 2018

From Amiram Eldar, Oct 10 2020: (Start)

Sum_{n>=1} 1/a(n) = zeta(3)/8 (A276712).

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/32. (End)