|
%I #53 Sep 08 2022 08:45:00
%S 0,1,4,28,160,976,5824,35008,209920,1259776,7558144,45349888,
%T 272097280,1632587776,9795518464,58773127168,352638730240,
%U 2115832446976,12694994550784,76169967566848,457019804876800,2742118830309376,16452712979759104
%N a(n) = (6^n - (-2)^n)/8.
%C The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - _Felix P. Muga II_, Mar 10 2014
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.1(b).
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 194-196.
%H Vincenzo Librandi, <a href="/A053524/b053524.txt">Table of n, a(n) for n = 0..1000</a>
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,12).
%F E.g.f.: (exp(6*x) - exp(-2*x))/8.
%F a(n) = 2^(n-3) * (3^n - (-1)^n) = 2^(n-3)*A105723(n).
%F a(n) = 4*a(n-1) + 12*a(n-2), with a(0)=0, a(1)=1.
%F G.f.: x / ((1+2*x)*(1-6*x)). - _Colin Barker_, Mar 11 2014
%p A053524:=n->(6^n-(-2)^n)/8; seq(A053524(n), n=0..30); # _Wesley Ivan Hurt_, Mar 11 2014
%t Table[(6^n -(-2)^n)/8, {n, 0, 30}] (* _Vincenzo Librandi_, Mar 11 2014 *)
%o (Sage) [lucas_number1(n,4,-12) for n in range(0, 30)] # _Zerinvary Lajos_, Apr 23 2009
%o (Magma) [2^n/8*(3^n-(-1)^n): n in [0..30]]; // _Vincenzo Librandi_, Mar 11 2014
%o (PARI) a(n) = (6^n-(-2)^n)/8; \\ _Joerg Arndt_, Mar 11 2014
%o (PARI) Vec(-x/((2*x+1)*(6*x-1)) + O(x^30)) \\ _Colin Barker_, Mar 11 2014
%Y Cf. A015518.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, _Barry E. Williams_, Jan 15 2000
|
