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a(n) = 2^n*(2^n - 1)*(2^n - 2)/6.
1

%I #28 Sep 08 2022 08:44:49

%S 0,0,4,56,560,4960,41664,341376,2763520,22238720,178433024,1429559296,

%T 11444858880,91592417280,732873539584,5863525154816,46910348656640,

%U 375291379056640,3002365391929344,24019060573863936,192153034345676800,1537226473786572800

%N a(n) = 2^n*(2^n - 1)*(2^n - 2)/6.

%H Vincenzo Librandi, <a href="/A026740/b026740.txt">Table of n, a(n) for n = 0..300</a>

%H J. Brandts and C. Cihangir, <a href="http://www.math.cas.cz/~am2013/proceedings/contributions/brandts.pdf">Counting triangles that share their vertices with the unit n-cube</a>, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (14,-56,64).

%F a(n) = binomial(2^n, 3).

%F a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3) for n>=3. - _Harvey P. Dale_, Jun 20 2012

%p seq(binomial(2^n, 3), n=0..20); # _Zerinvary Lajos_, Feb 22 2008

%t Binomial[2^Range[0,20],3] (* or *) LinearRecurrence[{14,-56,64},{0,0,4},21] (* _Harvey P. Dale_, Jun 20 2012 *)

%o (Magma) [2^n*(2^n-1)*(2^n-2)/6: n in [0..20] ]; // _Vincenzo Librandi_, May 23 2011

%o (PARI) vector(20, n, binomial(2^(n-1), 6) ) \\ _G. C. Greubel_, Oct 26 2019

%o (Sage) [binomial(2^n, 6) for n in (0..20)] # _G. C. Greubel_, Oct 26 2019

%o (GAP) List([0..20], n-> Binomial(2^n, 3) ); # _G. C. Greubel_, Oct 26 2019

%Y Cf. A000079, A006516.

%K nonn

%O 0,3

%A _N. J. A. Sloane_