cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A010469 Decimal expansion of square root of 12.

Original entry on oeis.org

3, 4, 6, 4, 1, 0, 1, 6, 1, 5, 1, 3, 7, 7, 5, 4, 5, 8, 7, 0, 5, 4, 8, 9, 2, 6, 8, 3, 0, 1, 1, 7, 4, 4, 7, 3, 3, 8, 8, 5, 6, 1, 0, 5, 0, 7, 6, 2, 0, 7, 6, 1, 2, 5, 6, 1, 1, 1, 6, 1, 3, 9, 5, 8, 9, 0, 3, 8, 6, 6, 0, 3, 3, 8, 1, 7, 6, 0, 0, 0, 7, 4, 1, 6, 2, 2, 9, 2, 3, 7, 3, 5, 1, 4, 4, 9, 7, 1, 5
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

3+sqrt(12) is the ratio of the radii of the three identical kissing circles to that of their inner Soddy circle. - Lekraj Beedassy, Mar 04 2006
sqrt(12)-3 = 2*sqrt(3)-3 is the area of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link). - Rick L. Shepherd, Jun 24 2006
Continued fraction expansion is 3 followed by {2, 6} repeated (A040008). - Harry J. Smith, Jun 02 2009
Surface of a regular octahedron with unit edge, and twice the surface of a regular tetrahedron with unit edge. - Stanislav Sykora, Nov 21 2013
Imaginary part of the square of a complex cubic root of 64 (real part is -2). - Alonso del Arte, Jan 13 2014

Examples

			3.4641016151377545870548926830...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.31.4 and 2.31.5, pp. 201-202.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, ยง12.4 Theorems and Formulas (Solid Geometry), p. 450.

Links

Crossrefs

Cf. A120683.
Cf. A040008 (continued fraction), A041016 (numerators of convergents), A041017 (denominators).
Cf. A002194 (surface of tetrahedron), A010527 (surface of icosahedron/10), A131595 (surface of dodecahedron).

Programs

  • Maple
    evalf[100](sqrt(12)); # Muniru A Asiru, Feb 12 2019
  • Mathematica
    RealDigits[N[Sqrt[12], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)
  • PARI
    default(realprecision, 20080); x=sqrt(12); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010469.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Formula

Equals 2*sqrt(3) = 2*A002194. - Rick L. Shepherd, Jun 24 2006

A041017 Denominators of continued fraction convergents to sqrt(12).

Original entry on oeis.org

1, 2, 13, 28, 181, 390, 2521, 5432, 35113, 75658, 489061, 1053780, 6811741, 14677262, 94875313, 204427888, 1321442641, 2847313170, 18405321661, 39657956492, 256353060613, 552364077718, 3570537526921
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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
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
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
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
Also, values i where A227790(i)/i reaches a new maximum (conjectured). - Ralf Stephan, Sep 23 2013

Links

Crossrefs

Cf. A010469, A040008, A041016 (numerators).

Programs

  • Maple
    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
  • Mathematica
    Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[12],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Denominator[Convergents[Sqrt[12],50]] (* Harvey P. Dale, Feb 18 2012 *)
a0[n_] := ((7-4*Sqrt[3])^n*(2+Sqrt[3])-(-2+Sqrt[3])*(7+4*Sqrt[3])^n)/4 // Simplify
a1[n_] := 2*Sum[a0[i], {i, 1, n}]
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[11]]] (* Gerry Martens, Jul 10 2015 *)

Formula

G.f.: (1+2*x-x^2)/(1-14*x^2+x^4). - Colin Barker, Jan 01 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)]:
a0(n) = ((7-4*sqrt(3))^n*(2+sqrt(3)) - (-2+sqrt(3))*(7+4*sqrt(3))^n)/4.
a1(n) = 2*Sum_{i=1..n} a0(i). (End)

A040032 Continued fraction for sqrt(39).

Original entry on oeis.org

6, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			6.2449979983983982058468931... = 6 + 1/(4 + 1/(12 + 1/(4 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 05 2009

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Links

Crossrefs

Cf. A010493 (decimal expansion).

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[39],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},100,{12,4}] (* or *) Join[{6},LinearRecurrence[{0,1},{4,12},100]] (* Harvey P. Dale, Feb 09 2015 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 35000); x=contfrac(sqrt(39)); for (n=0, 20000, write("b040032.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

Formula

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 2*A040008(n).
G.f.: 2*(3 + 2*x + 3*x^2)/(1 - x^2). (End)

A064849 Period of continued fraction for sqrt(3)*n.

Original entry on oeis.org

2, 2, 2, 2, 4, 8, 2, 4, 10, 4, 2, 8, 6, 2, 2, 8, 6, 20, 4, 8, 14, 8, 8, 12, 16, 2, 30, 2, 16, 4, 18, 16, 10, 20, 16, 20, 18, 8, 6, 12, 2, 12, 8, 8, 6, 20, 20, 20, 30, 36, 6, 2, 8, 68, 14, 2, 16, 32, 22, 4, 38, 18, 40, 36, 6, 28, 10, 20, 40, 8, 4, 40, 18, 22, 16, 12, 28, 2, 46, 20, 98, 8
Offset: 1

Views

Author

R. K. Guy, Oct 26 2001

Keywords

Examples

			A040001 (cfrac for n=1) has period length 2, so a(1)=2. A040008 (cfrac for n=2) has period length 2, so a(2)=2. A040021 (cfrac for =3) has period length 2, so a(3)=2. - _R. J. Mathar_, Feb 10 2016

Programs

  • Mathematica
    Table[Length[Last[ContinuedFraction[Sqrt[3] n]]], {n, 128}]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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