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%I A064200 #39 Feb 22 2023 01:50:00
%S A064200 0,0,24,72,144,240,360,504,672,864,1080,1320,1584,1872,2184,2520,2880,
%T A064200 3264,3672,4104,4560,5040,5544,6072,6624,7200,7800,8424,9072,9744,
%U A064200 10440,11160,11904,12672,13464,14280,15120,15984,16872,17784,18720,19680,20664,21672
%N A064200 a(n) = 12*n*(n-1).
%D A064200 Luigi Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Band III_2, Heft 3, Leipzig: B. G. Teubner, 1906, p. 341.
%H A064200 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A064200 Leo Tavares, <a href="/A064200/a064200.jpg">Illustration: Twin Stars</a>.
%H A064200 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A064200 a(n) = 24*(n-1) + a(n-1) for n>0, with a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F A064200 a(0)=0, a(1)=0, a(2)=24, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - _Harvey P. Dale_, Jul 22 2015
%F A064200 G.f.: -(24*x^2)/(x-1)^3. - _Harvey P. Dale_, Jul 22 2015
%F A064200 a(n) = 2*A003154(n) - 2. See Twin Stars illustration. - _Leo Tavares_, Aug 23 2021
%F A064200 From _Amiram Eldar_, Feb 22 2023: (Start)
%F A064200 Sum_{n>=2} 1/a(n) = 1/12.
%F A064200 Sum_{n>=2} (-1)^n/a(n) = (2*log(2) - 1)/12.
%F A064200 Product_{n>=2} (1 - 1/a(n)) = -(12/Pi)*cos(Pi/sqrt(3)).
%F A064200 Product_{n>=2} (1 + 1/a(n)) = (12/Pi)*cos(Pi/sqrt(6)). (End)
%t A064200 Table[12n(n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,0,24},40] (* _Harvey P. Dale_, Jul 22 2015 *)
%t A064200 Join[{0},24*Accumulate[Range[0,60]]] (* _Harvey P. Dale_, Dec 17 2022 *)
%o A064200 (PARI) a(n)=12*n*(n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A064200 Cf. A262221, A003154.
%K A064200 nonn,easy
%O A064200 0,3
%A A064200 Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Sep 22 2001

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