A144536 Denominators of continued fraction convergents to sqrt(3)/2.

1, 1, 7, 15, 97, 209, 1351, 2911, 18817, 40545, 262087, 564719, 3650401, 7865521, 50843527, 109552575, 708158977, 1525870529, 9863382151, 21252634831, 137379191137, 296011017105, 1913445293767, 4122901604639, 26650854921601, 57424611447841, 371198523608647

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

John Elias, Double-Snowflake Fraction

Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Formula

G.f.: (1 + x - 7*x^2 + x^3)/(1 - 14*x^2 + x^4). - Colin Barker, Jan 01 2012

a(n) = 14*a(n-2) - a(n-4). - Sergei N. Gladkovskii, Jun 07 2015

a(n) = ((3+sqrt(3))*((-2+sqrt(3))^n + (2+sqrt(3))^n) - (-3+sqrt(3))*((-2-sqrt(3))^n + (2-sqrt(3))^n))/12. - Vaclav Kotesovec, Jun 08 2015

From John Elias, Dec 02 2021: (Start)

a(2*n) = 6*A001353(n)^2 + 1. See illustration in links.

a(2*n+1) = 2*a(2*n) + a(2*n-1). (End)

Example

0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...

Maple

with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2, confrac); [seq(nthconver(cf, i), i=0..100)];

Mathematica

Denominator[Convergents [Sqrt[3]/2, 30]] (* Vincenzo Librandi, Feb 01 2014 *)
LinearRecurrence[{0, 14, 0, -1}, {1, 1, 7, 15}, 30] (* Harvey P. Dale, Sep 15 2017 *)

Crossrefs

Cf. A126664, A144535, A002531/A002530.

Bisections give A011943, A028230.

Sequence in context: A335758 A343279 A041413 * A231399 A231466 A032004

Adjacent sequences: A144533 A144534 A144535 * A144537 A144538 A144539

Keyword

nonn,easy,frac

Author

N. J. A. Sloane, Dec 29 2008

Status

approved