This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A041017 #52 Aug 10 2024 21:38:27
%S A041017 1,2,13,28,181,390,2521,5432,35113,75658,489061,1053780,6811741,
%T A041017 14677262,94875313,204427888,1321442641,2847313170,18405321661,
%U A041017 39657956492,256353060613,552364077718,3570537526921
%N A041017 Denominators of continued fraction convergents to sqrt(12).
%C A041017 a(2n+1)/a(2n) tends to 1/(sqrt(12) - 3) = 2.154700538...; e.g., a(7)/a(6) = 5432/2521 = 2.1547005...; but a(2n)/a(2n - 1) tends to 6.464101615... = sqrt(12) + 3; e.g., a(8)/a(7) = 35113/5432 = 6.46101620... - _Gary W. Adamson_, Mar 28 2004
%C A041017 The constant sqrt(12) + 3 = 6.464101615... is the "curvature" (reciprocal of the radius) of the inner or 4th circle in the Descartes circle equation; given 3 mutually tangent circles of radius 1, the radius of the innermost tangential circle = 0.1547005383... = 1/(sqrt(12) + 3). The Descartes circle equation states that given 4 mutually tangent circles (i.e., 3 tangential plus the innermost circle) with curvatures a,b,c,d (curvature = 1/r), then (a^2 + b^2 + c^2 + d^2) = 1/2(a + b + c + d)^2. - _Gary W. Adamson_, Mar 28 2004
%C A041017 Sequence also gives numerators in convergents to barover[6,2] = CF: [6,2,6,2,6,2,...] = 0.1547005... = 1/(sqrt(12) + 3), the first few convergents being 1/6, 2/13, 13/84, 28/181, 181/1170, 390/2521... with 390/2521 = 0.154700515... - _Gary W. Adamson_, Mar 28 2004
%C A041017 Sqrt(12) = 3 + continued fraction [2, 6, 2, 6, 2, 6, ...] = 6/2 + 6/13 + 6/(13*181) + 6/(181*2521) + ... - _Gary W. Adamson_, Dec 21 2007
%C A041017 Also, values i where A227790(i)/i reaches a new maximum (conjectured). - _Ralf Stephan_, Sep 23 2013
%H A041017 Vincenzo Librandi, <a href="/A041017/b041017.txt">Table of n, a(n) for n = 0..200</a>
%H A041017 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,14,0,-1).
%F A041017 G.f.: (1+2*x-x^2)/(1-14*x^2+x^4). - _Colin Barker_, Jan 01 2012
%F A041017 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041017 Interspersion of 2 sequences [a0(n),a1(n)]:
%F A041017 a0(n) = ((7-4*sqrt(3))^n*(2+sqrt(3)) - (-2+sqrt(3))*(7+4*sqrt(3))^n)/4.
%F A041017 a1(n) = 2*Sum_{i=1..n} a0(i). (End)
%p A041017 with (numtheory): seq( nthdenom(cfrac(sin(Pi/6)*tan(Pi/3),25),i)-nthnumer(cfrac(sin(Pi/6)*tan(Pi/3),25),i), i=2..24 ); # _Zerinvary Lajos_, Feb 10 2007
%t A041017 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[12],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%t A041017 Denominator[Convergents[Sqrt[12],50]] (* _Harvey P. Dale_, Feb 18 2012 *)
%t A041017 a0[n_] := ((7-4*Sqrt[3])^n*(2+Sqrt[3])-(-2+Sqrt[3])*(7+4*Sqrt[3])^n)/4 // Simplify
%t A041017 a1[n_] := 2*Sum[a0[i], {i, 1, n}]
%t A041017 Flatten[MapIndexed[{a0[#],a1[#]}&,Range[11]]] (* _Gerry Martens_, Jul 10 2015 *)
%Y A041017 Cf. A010469, A040008, A041016 (numerators).
%K A041017 nonn,cofr,frac,easy
%O A041017 0,2
%A A041017 _N. J. A. Sloane_