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

%I A016743 #63 Oct 19 2024 15:57:32
%S A016743 0,8,64,216,512,1000,1728,2744,4096,5832,8000,10648,13824,17576,21952,
%T A016743 27000,32768,39304,46656,54872,64000,74088,85184,97336,110592,125000,
%U A016743 140608,157464,175616,195112,216000,238328,262144,287496,314432
%N A016743 Even cubes: a(n) = (2*n)^3.
%C A016743 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
%H A016743 Vincenzo Librandi, <a href="/A016743/b016743.txt">Table of n, a(n) for n = 0..10000</a>
%H A016743 Hilko Koning, <a href="http://www.hilko.net/216.jpg">216</a> neodymium magnets for n=3.
%H A016743 Ana Rechtman, <a href="http://images-archive.math.cnrs.fr/Mars-2022-1er-defi.html">Mars 2022, 1er défi</a>, Images des Mathématiques, CNRS, 2022 (in French).
%H A016743 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A016743 a(n) = (2*n)^3 = 8*n^3.
%F A016743 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
%F A016743 E.g.f.: 8*x*(1 +3*x +x^2)*exp(x). - _G. C. Greubel_, Sep 15 2018
%F A016743 From _Amiram Eldar_, Oct 10 2020: (Start)
%F A016743 Sum_{n>=1} 1/a(n) = zeta(3)/8 (A276712).
%F A016743 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/32. (End)
%p A016743 A016743:=n->(2*n)^3: seq(A016743(n), n=0..50); # _Wesley Ivan Hurt_, Sep 15 2018
%t A016743 Range[0, 78, 2]^3 (* _Alonso del Arte_, Apr 06 2013 *)
%o A016743 (Magma) [(2*n)^3: n in [0..50]]; // _Vincenzo Librandi_, Sep 05 2011
%o A016743 (PARI) a(n) = 8*n^3; \\ _Joerg Arndt_, Apr 07 2013
%Y A016743 Even bisection of A000578, cf. A016755.
%Y A016743 Cf. A016803 (even bisection), A016827 (odd bisection), A033581, A276712.
%K A016743 nonn,easy
%O A016743 0,2
%A A016743 _N. J. A. Sloane_