%I #27 Sep 08 2022 08:45:01

%S 1,8,31,116,433,1616,6031,22508,84001,313496,1169983,4366436,16295761,

%T 60816608,226970671,847066076,3161293633,11798108456,44031140191,

%U 164326452308,613274669041,2288772223856

%N a(n) = 4*a(n-1) - a(n-2) with a(0)=1, a(1)=8.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H G. C. Greubel, <a href="/A055845/b055845.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 (4,-1).

%F a(n) = (8*((2+sqrt(3))^n - (2-sqrt(3))^n) - ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/(2*sqrt(3)).

%F G.f.: (1+4*x)/(1-4*x+x^2).

%F a(n)^2 = 3*A144721(n)^2 - 11. - _Sture Sjöstedt_, Nov 30 2011

%F From _G. C. Greubel_, Jan 20 2020: (Start)

%F a(n) = ChebyshevU(n,2) + 4*ChebyshevU(n-1,2).

%F E.g.f.: exp(2*x)*( cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) ). (End)

%p seq( simplify(ChebyshevU(n,2) + 4*ChebyshevU(n-1,2)), n=0..30); # _G. C. Greubel_, Jan 20 2020

%t LinearRecurrence[{4,-1}, {1,8}, 30] (* _Sture Sjöstedt_, Nov 30 2011 *)

%t Table[ChebyshevU[n, 2] + 4*ChebyshevU[n-1, 2], {n,0,30}] (* _G. C. Greubel_, Jan 20 2020 *)

%o (PARI) a(n) = polchebyshev(n,2,2) + 4*polchebyshev(n-1,2,2); \\ _G. C. Greubel_, Jan 20 2020

%o (Magma) I:=[1,8]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 20 2020

%o (Sage) [chebyshev_U(n,2) +4*chebyshev_U(n-1,2) for n in (0..30)] # _G. C. Greubel_, Jan 20 2020

%o (GAP) a:=[1,8];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 20 2020

%Y Cf. A054485.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, May 31 2000