A136325 a(n) = 8*a(n-1)-a(n-2) with a(0)=0 and a(1)=3.

0, 3, 24, 189, 1488, 11715, 92232, 726141, 5716896, 45009027, 354355320, 2789833533, 21964312944, 172924670019, 1361433047208, 10718539707645, 84386884613952, 664376537203971, 5230625413017816, 41180626766938557

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

Table of n, a(n) for n=0..19.

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