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%I #44 Dec 18 2021 15:06:08
%S 1,1,7,15,97,209,1351,2911,18817,40545,262087,564719,3650401,7865521,
%T 50843527,109552575,708158977,1525870529,9863382151,21252634831,
%U 137379191137,296011017105,1913445293767,4122901604639,26650854921601,57424611447841,371198523608647
%N Denominators of continued fraction convergents to sqrt(3)/2.
%H Vincenzo Librandi, <a href="/A144536/b144536.txt">Table of n, a(n) for n = 0..200</a>
%H John Elias, <a href="/A144536/a144536.png">Double-Snowflake Fraction</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,14,0,-1).
%F G.f.: (1 + x - 7*x^2 + x^3)/(1 - 14*x^2 + x^4). - _Colin Barker_, Jan 01 2012
%F a(n) = 14*a(n-2) - a(n-4). - _Sergei N. Gladkovskii_, Jun 07 2015
%F 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
%F From _John Elias_, Dec 02 2021: (Start)
%F a(2*n) = 6*A001353(n)^2 + 1. See illustration in links.
%F a(2*n+1) = 2*a(2*n) + a(2*n-1). (End)
%e 0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...
%p with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2,confrac); [seq(nthconver(cf,i), i=0..100)];
%t Denominator[Convergents [Sqrt[3]/2, 30]] (* _Vincenzo Librandi_, Feb 01 2014 *)
%t LinearRecurrence[{0,14,0,-1},{1,1,7,15},30] (* _Harvey P. Dale_, Sep 15 2017 *)
%Y Cf. A126664, A144535, A002531/A002530.
%Y Bisections give A011943, A028230.
%K nonn,easy,frac
%O 0,3
%A _N. J. A. Sloane_, Dec 29 2008
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