|
%I #27 Sep 08 2022 08:45:15
%S 4,22,128,746,4348,25342,147704,860882,5017588,29244646,170450288,
%T 993457082,5790292204,33748296142,196699484648,1146448611746,
%U 6681992185828,38945504503222,226991034833504,1323000704497802,7711013192153308,44943078448422046
%N Bisection of A048654.
%H Colin Barker, <a href="/A100525/b100525.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F G.f.: 2*(2-x)/(1-6*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F a(0)=4, a(1)=22, a(n) = 6*a(n-1) - a(n-2) for n>1. - _Philippe Deléham_, Sep 19 2009
%F a(n) = 2*A038725(n+1). - _R. J. Mathar_, Sep 27 2014
%F a(n) = ( (5 + 4*sqrt(2))*(3 + 2*sqrt(2))^n - (5 - 4*sqrt(2))*(3 - 2*sqrt(2))^n )/(2*sqrt(2)). - _Colin Barker_, Oct 13 2015
%F From _G. C. Greubel_, Jun 28 2022: (Start)
%F a(n) = 2*( 2*ChebyshevU(n, 3) - ChenyshevU(n-1, 3) ).
%F E.g.f.: 2*exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (5/(2*sqrt(2)))*sinh(2*sqrt(2)*x) ). (End)
%t CoefficientList[Series[(4-2x)/(1-6x+x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Oct 13 2015 *)
%t LinearRecurrence[{6,-1},{4,22},30] (* _Harvey P. Dale_, Mar 25 2016 *)
%o (PARI) Vec((4-2*x)/(1-6*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 13 2015
%o (Magma) I:=[4,22,128]; [n le 3 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Oct 13 2015
%o (SageMath) [2*(2*chebyshev_U(n,3) - chebyshev_U(n-1,3)) for n in (0..30)] # _G. C. Greubel_, Jun 28 2022
%Y Cf. A001109, A038761, A048654.
%K easy,nonn
%O 0,1
%A Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 24 2004
|
