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.

A010524 Decimal expansion of square root of 72.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

This is also the ratio of the volume of a cube to the volume of a regular tetrahedron of the same edge length. - Rick L. Shepherd, May 29 2002
Continued fraction expansion is 8 followed by {2, 16} repeated. - Harry J. Smith, Jun 08 2009
Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(8,15,17), see A195284. - Clark Kimberling, Sep 14 2011
Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(7,24,25). - Clark Kimberling, Sep 14 2011

Examples

			8.485281374238570292810132345258188471418031252261688439060078427944394...

Links

Crossrefs

Cf. A040063 (continued fraction), A002193, A195284.

Programs

  • Mathematica
    RealDigits[N[72^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)
  • PARI
    default(realprecision, 20080); x=sqrt(72); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010524.txt", n, " ", d));  \\ Harry J. Smith, Jun 08 2009

Formula

Equals 6*A002193. - Omar E. Pol, Mar 09 2021

A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 5, 3, 1, 2, 3, 3, 6, 5, 3, 1, 2, 3, 7, 3, 7, 7, 5, 3, 1, 2, 3, 5, 6, 3, 9, 5, 5, 5, 3, 1, 2, 3, 3, 3, 4, 3, 11, 9, 7, 13, 5, 3, 1, 2, 3, 7, 6, 7, 5, 3, 7, 8, 7, 5, 12, 5, 3, 1, 2, 3, 11, 3, 9, 7, 9, 3, 8, 6, 5, 13, 7, 5, 5, 3, 1, 2, 3, 3, 6, 11, 3, 7, 6, 3, 9, 9, 11, 17, 5, 5, 12, 5
Offset: 1

Views

Author

Frank Ellermann, Feb 23 2002

Keywords

Examples

			a(2)=2: [1,(2)+ ]; a(3)=3: [1,(1,2)+ ]; a(4)=1: [2]; a(5)=2: [2,(4)+ ].

References

  • H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, table 1.

Crossrefs

Related sequences: 2 : A040000, ..., 44: A040037, 48: A040041, ..., 51: A040043, 56: A040048, 60: A040052, 63: A040055, ..., 66: A040057. 68: A040059, 72: A040063, 80: A040071.
Related sequences: 45: A010135, ..., 47: A010137, 52: A010138, ..., 55: A010141, 57: A010142, ..., 59: A010144. 61: A010145, 62: A010146. 67: A010147, 69: A010148, ..., 71: A010150.
Cf. A003285.

Programs

  • Python
    from sympy import continued_fraction_periodic
def A067280(n): return len((a := continued_fraction_periodic(0,1,n))[:1]+(a[1] if a[1:] else [])) # Chai Wah Wu, Jun 14 2022

Formula

a(n) = A003285(n) + 1. - Andrey Zabolotskiy, Jun 23 2020

Extensions

Name clarified by Michel Marcus, Jun 22 2020

A041127 Denominators of continued fraction convergents to sqrt(72).

Original entry on oeis.org

1, 2, 33, 68, 1121, 2310, 38081, 78472, 1293633, 2665738, 43945441, 90556620, 1492851361, 3076259342, 50713000833, 104502261008, 1722749176961, 3550000614930, 58522759015841, 120595518646612, 1988051057361633, 4096697633369878, 67535213191279681
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041126, A040063, A020829, A010524.

Programs

  • Magma
    I:=[1, 2, 33, 68]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013
  • Mathematica
    Denominator/@Convergents[Sqrt[72], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + 2 x - x^2)/(x^4 - 34 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
a0[n_] := ((3+2*Sqrt[2])/(17+12*Sqrt[2])^n+(3-2*Sqrt[2])*(17+ 12*Sqrt[2])^n)/6 // Simplify
a1[n_] := (-1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/(12*Sqrt[2]) // FullSimplify
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[20]]] (* Gerry Martens, Jul 10 2015 *)
  • PARI
    a(n)=my(v=contfrac(sqrt(72),n),s=v[n]);forstep(k=n-1,1,-1,s=v[k]+1/s);denominator(s) \\ Charles R Greathouse IV, Jul 05 2011

Formula

G.f.: -(x^2-2*x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 13 2013
a(n) = 34*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 11 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((3+2*sqrt(2))/(17+12*sqrt(2))^n+(3-2*sqrt(2))*(17+12*sqrt(2))^n)/6.
a1(n) = (-1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/(12*sqrt(2)). (End)

A144532 Continued fraction for sqrt(8/9).

Original entry on oeis.org

0, 1, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16
Offset: 0

Views

Author

N. J. A. Sloane, Dec 29 2008

Keywords

Comments

Or, continued fraction for 2*sqrt(2)/3.

Crossrefs

Cf. A040000, A144533/A144534.
Essentially the same as A040063.
Showing 1-4 of 4 results.

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

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