|
%I #35 Sep 23 2019 14:11:47
%S 0,3,24,189,1488,11715,92232,726141,5716896,45009027,354355320,
%T 2789833533,21964312944,172924670019,1361433047208,10718539707645,
%U 84386884613952,664376537203971,5230625413017816,41180626766938557
%N a(n) = 8*a(n-1)-a(n-2) with a(0)=0 and a(1)=3.
%C Nonnegative integers k such that 15*k^2 + 9 is a square.
%C From the recurrence we have a(n) = sqrt(15)*((4 + sqrt(15))^n - (4 - sqrt(15))^n)/10.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-1).
%F From _Colin Barker_, Jan 24 2013: (Start)
%F a(n) = (sqrt(3/5)*(-(4-sqrt(15))^n + (4+sqrt(15))^n))/2.
%F G.f.: 3*x/(x^2-8*x+1). (End)
%F a(n) = 3 * A001090(n).
%F For n > 0, a(n) is the denominator of the continued fraction [2,3,2,3,...,2,3] with n repetitions of 2,3. For the numerators see A070997. - _Greg Dresden_, Sep 12 2019
%e G.f. = 3*x + 24*x^2 + 189*x^3 + 1488*x^4 + 11715*x^5 + 92232*x^6 + 726141*x^7 + ...
%t Do[If[IntegerQ[Sqrt[3 (3 + 5 x^2)]], Print[{x, Sqrt[3 (3 + 5 x^2)]}]], {x, 0, 2000000}]
%t LinearRecurrence[{8,-1},{0,3},30] (* _Harvey P. Dale_, Aug 18 2014 *)
%t a[ n_] := 3 ChebyshevU[ n - 1, 4]; (* _Michael Somos_, Oct 14 2015 *)
%t a[ n_] := 3/2 ((4 + Sqrt[15])^n - (4 - Sqrt[15])^n) / Sqrt[15] // Simplify; (* _Michael Somos_, Oct 14 2015 *)
%o (PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 5}; /* _Michael Somos_, Apr 05 2008 */
%o (PARI) {a(n) = 3 * polchebyshev(n-1, 2, 4)}; /* _Michael Somos_, Oct 14 2015 */
%o (PARI) {a(n) = 3 * imag( (4 + quadgen(60))^n )}; /* _Michael Somos_, Oct 14 2015 */
%Y Cf. A001090.
%K nonn,easy
%O 0,2
%A _Lorenz H. Menke, Jr._, Mar 26 2008
%E Definition corrected by _Bruno Berselli_, Jan 24 2013
%E Definition, comments, formulas further corrected by _Greg Dresden_, Sep 13 2019
%E Exchanged definition and comment, in order to retain offset 0. - _N. J. A. Sloane_, Sep 23 2019
| 