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 A136325 a(n) = 8*a(n-1)-a(n-2) with a(0)=0 and a(1)=3. 3
 0, 3, 24, 189, 1488, 11715, 92232, 726141, 5716896, 45009027, 354355320, 2789833533, 21964312944, 172924670019, 1361433047208, 10718539707645, 84386884613952, 664376537203971, 5230625413017816, 41180626766938557 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Nonnegative integers k such that 15*k^2 + 9 is a square. From the recurrence we have a(n) = sqrt(15)*((4 + sqrt(15))^n - (4 - sqrt(15))^n)/10. LINKS Index entries for linear recurrences with constant coefficients, signature (8,-1). FORMULA From Colin Barker, Jan 24 2013: (Start) a(n) = (sqrt(3/5)*(-(4-sqrt(15))^n + (4+sqrt(15))^n))/2. G.f.: 3*x/(x^2-8*x+1). (End) a(n) = 3 * A001090(n). 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 EXAMPLE G.f. = 3*x + 24*x^2 + 189*x^3 + 1488*x^4 + 11715*x^5 + 92232*x^6 + 726141*x^7 + ... MATHEMATICA Do[If[IntegerQ[Sqrt[3 (3 + 5 x^2)]], Print[{x, Sqrt[3 (3 + 5 x^2)]}]], {x, 0, 2000000}] LinearRecurrence[{8, -1}, {0, 3}, 30] (* Harvey P. Dale, Aug 18 2014 *) a[ n_] := 3 ChebyshevU[ n - 1, 4]; (* Michael Somos, Oct 14 2015 *) a[ n_] := 3/2 ((4 + Sqrt[15])^n - (4 - Sqrt[15])^n) / Sqrt[15] // Simplify; (* Michael Somos, Oct 14 2015 *) PROG (PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 5}; /* Michael Somos, Apr 05 2008 */ (PARI) {a(n) = 3 * polchebyshev(n-1, 2, 4)}; /* Michael Somos, Oct 14 2015 */ (PARI) {a(n) = 3 * imag( (4 + quadgen(60))^n )}; /* Michael Somos, Oct 14 2015 */ CROSSREFS Cf. A001090. Sequence in context: A213100 A027324 A122741 * A194888 A103333 A037762 Adjacent sequences:  A136322 A136323 A136324 * A136326 A136327 A136328 KEYWORD nonn,easy AUTHOR Lorenz H. Menke, Jr., Mar 26 2008 EXTENSIONS Definition corrected by Bruno Berselli, Jan 24 2013 Definition, comments, formulas further corrected by Greg Dresden, Sep 13 2019 Exchanged definition and comment, in order to retain offset 0. - N. J. A. Sloane, Sep 23 2019 STATUS approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)