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
Examples
0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...
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).
Programs
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Maple
with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2,confrac); [seq(nthconver(cf,i), i=0..100)];
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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 *)
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)