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 1-10 of 74 results. Next

A020806 Decimal expansion of 1/7.

Original entry on oeis.org

1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2
Offset: 0

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Comments

142857 and 999999 = 7*142857 are first and last Kaprekar numbers with six digits. Note a(n) + a(n+3) = 9. (142857^2 = 20408122449; 20408 + 122449 = 142857.) a(n)^2 = 1, 16, 4, 64, 25, 49, ... - Paul Curtz, Aug 24 2009
The constant 19 + 1/7 = 19.142857... is the Kirchhoff index of the Möbius ladder graph on v=8 vertices. The Laplacian matrix has the eigenvalues 4 (one time), 4-sqrt(2) (2 times), 4+sqrt(2) (2 times), 2 (2 times) and 0 (one time). Then the Kirchhoff index is v times the sum over the inverse, nonzero eigenvalues. - R. J. Mathar, Feb 13 2011
Decimal expansion of -99*(zeta(-5) + zeta(-9)) - 1. - Arkadiusz Wesolowski, Sep 15 2013
Also, decimal expansion of Sum_{i>0} 1/8^i. - Bruno Berselli, Jan 03 2014
The points whose coordinates are overlapping pairs of digits of this sequence, (1, 4), (4, 2), (2, 8), (8, 5), (5, 7) and (7, 1), all lie on one ellipse, with equation 19*x^2 + 36*x*y + 41*y^2 - 333*x - 531*y = -1638. Overlapping pairs of pairs of digits, (14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42), also yield 6 points on one ellipse, with equation -165104*x^2 + 160804*x*y + 8385498*x - 41651*y^2 - 3836349*y = 7999600. (See book by Wells and MathWorld link.) - M. F. Hasler, Oct 25 2017

Examples

			0.142857142857142857...
		

References

  • H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrüche'.
  • D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

Crossrefs

Programs

  • Magma
    I:=[1,4,2,8]; [n le 4 select I[n] else Self(n-1)-Self(n-3)+Self(n-4): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015
    
  • Maple
    Digits:=100: evalf(1/7); # Wesley Ivan Hurt, Jun 28 2016
  • Mathematica
    CoefficientList[Series[(1 + 3 x - 2 x^2 + 7 x^3) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
    realDigitsRecip[7] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Sep 18 2024 *)
  • PARI
    1/7. \\ Charles R Greathouse IV, Sep 24 2015
    
  • PARI
    digits(10^99\7) \\ M. F. Hasler, Oct 25 2017

Formula

From Reinhard Zumkeller, Oct 06 2008: (Start)
A028416(1)=7; A002371(A049084(7)) = A002371(4) = 6.
a(n+6) = a(n), a(n+6/2) = 9 - a(n). (End)
From Colin Barker, Aug 14 2012: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: (1+3*x-2*x^2+7*x^3) / ((1-x)*(1+x)*(1-x+x^2)). (End)
a(n) = A068028(n+2). - Zak Seidov, Mar 26 2015
a(n) = (27 - 11*cos(n*Pi) - 10*cos(n*Pi/3) - 6*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 28 2016
E.g.f.: (8*cosh(x) - exp(x/2)*(5*cos(sqrt(3)*x/2) + 3*sqrt(3)*sin(sqrt(3)*x/2)) + 19*sinh(x))/3. - Stefano Spezia, Dec 07 2024

A020765 Decimal expansion of 1/sqrt(8).

Original entry on oeis.org

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

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Multiplied by 10, this is the real and the imaginary part of sqrt(25i). - Alonso del Arte, Jan 11 2013
Radius of the midsphere (tangent to the edges) in a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013
The side of the largest cubical present that can be wrapped (with cutting) by a unit square of wrapping paper. See Problem 10716 link. - Michel Marcus, Jul 24 2018
The ratio between the thickness and diameter of a geometrically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on comparing the areal projections of the faces and sides of the coin on a circumscribing sphere. (Mosteller, 1965). See A020760 for a physical solution. - Amiram Eldar, Sep 01 2020

Examples

			1/sqrt(8) = 0.353553390593273762200422181052424519642417968844237018294...
		

References

  • Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 38, pp. 10 and 58-60.

Crossrefs

Cf. Midsphere radii in Platonic solids:
A020761 (octahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron).

Programs

Formula

A010503 divided by 2.
Equals A201488 minus 1/2. Equals 1/(A010487-4) minus 1/4. - Jon E. Schoenfield, Jan 09 2017
Equals Integral_{x=0..oo} x*exp(-x)*BesselJ(0,x) dx. - Kritsada Moomuang, Jun 03 2025

A020784 Decimal expansion of 1/sqrt(27).

Original entry on oeis.org

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

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Comments

This is the minimum ripple factor for a third-order Chebyshev filter for which the generalized reflectionless topology needs no negative elements. - Matthew A. Morgan, Oct 18 2017

Examples

			0.1924500897298752548363829268339858185492005837567089586728674....
		

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.16, pp. 495, 527.

Crossrefs

Programs

Formula

Equals Sum_{k>=0} binomial(2*k,k) * k/16^k. - Amiram Eldar, Aug 02 2020
Equals sqrt(3)/9. - Stefano Spezia, Dec 24 2024
Equals 1/A010482 = A020760/3 = sqrt(A021031) = A073010/Pi = A212886/2. - Hugo Pfoertner, Dec 24 2024

A092742 Decimal expansion of 1/Pi^2.

Original entry on oeis.org

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

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Author

Mohammad K. Azarian, Apr 12 2004

Keywords

Comments

The asymptotic density of squarefree numbers that are divisible by 5. - Amiram Eldar, Mar 25 2021

Examples

			0.101321183642337771443879463209727638904358774672246548845609...
		

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.6.1, p. 220.

Crossrefs

Cf. A000796 (Pi), A002388 (Pi^2), A091925 (Pi^3), A092425 (Pi^4), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8).
Cf. A049541 (1/Pi), A092743 (1/Pi^3), A092744 (1/Pi^4), A092745 (1/Pi^5), A092746 (1/Pi^6), A092747 (1/Pi^7), A092748 (1/Pi^8).

Programs

A020763 Decimal expansion of 1/sqrt(6).

Original entry on oeis.org

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

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Author

Keywords

Comments

Radius of the inscribed sphere (tangent to all faces) in a regular octahedron with unit edge. - Stanislav Sykora, Nov 21 2013

Examples

			0.408248290463863016366214012450981898660991246776111688072115427875...
		

References

  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Crossrefs

Cf. Platonic solids in radii: A020781 (tetrahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

Programs

Formula

From Michal Paulovic, Dec 09 2022: (Start)
Equals A157697/2 = A010503 * A020760 = 1/A010464.
Equals [0, 2; 2, 4] (periodic continued fraction expansion). (End)

A021083 Decimal expansion of 1/79.

Original entry on oeis.org

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

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Comments

The 13-digit cycle 1, 0, 1, 2, 6, 5, 8, 2, 2, 7, 8, 4, 8 in this sequence and the others based on seventy-ninths, gives the successive digits of the smallest integer which is multiplied by eight when the final digit is moved from the right hand end to the left hand end. - Ian Duff, Jan 09 2009

Examples

			1/79 = 0.01265822784810126582278481...
		

Programs

A113011 Decimal expansion of 1/(sqrt(e) - 1).

Original entry on oeis.org

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

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Author

Eric W. Weisstein, following a suggestion of Grover W. Hughes, Oct 09 2005

Keywords

Comments

Has continued fraction 1+2/(3+4/(5+6/7+...)).
Simple continued fraction is 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, {1, 4k+1, 1}, ..., . - Robert G. Wilson v, Jul 01 2007

Examples

			1.54149408253679828413110344447251463834045923684188210947413695663...
		

Crossrefs

Programs

  • Magma
    1/(Sqrt(Exp(1)) - 1); // G. C. Greubel, Apr 09 2018
  • Mathematica
    First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* Robert G. Wilson v, Jul 01 2007 *)
    f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; RealDigits[ f[61], 10, 105][[1]] (* Robert G. Wilson v, Jul 07 2012 *)
    Rest[realDigitsRecip[Sqrt[E]-1]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Nov 04 2024 *)
  • PARI
    1/(sqrt(exp(1)) - 1) \\ G. C. Greubel, Apr 09 2018
    

Formula

Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - Peter Bala, Oct 09 2013
Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - Antonio Graciá Llorente, Jan 21 2024
Conjecture: 1/(sqrt(e) - 1) = 1 + K_{n>=1} 2*n/(4*n^2-1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 1/(sqrt(e) - 1) = 1 + 2/3/(1 + 4/15/(1 + 6/35/(1 + ...))) (see Beit-Halachmi and Kaminer). - Stefano Spezia, Dec 27 2024
Equals 1/(A019774 - 1). - Hugo Pfoertner, Dec 27 2024

Extensions

Simpler definition from T. D. Noe, Oct 09 2005
Euler reference from David L. Harden, Oct 09 2005

A353444 Integers m such that the decimal expansion of 1/m contains the digit 8.

Original entry on oeis.org

7, 12, 14, 17, 19, 23, 26, 28, 29, 31, 34, 35, 38, 42, 43, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 102, 103, 104, 105, 107, 109, 112, 113, 114, 115, 116, 117, 118
Offset: 1

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Author

Keywords

Comments

If m is a term, 10*m is also a term, so terms with no trailing zeros are all primitive terms.

Examples

			m = 12 is a term since 1/12 = 0.08333333333...
m = 17 is a term since 1/17 = 0.05882352941176470588235294117647...
m = 125 is a term since 1/125 = 0.008.
		

Crossrefs

A351474 (largest digit=8) and A352161 (smallest digit=8) are subsequences.
Similar with digit k: A352154 (k=0), A353437 (k=1), A353438 (k=2), A353439 (k=3), A353440 (k=4), A353441 (k=5), A353442 (k=6), A353443 (k=7), this sequence (k=8), A333237 (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 150, MemberQ[f@#, 8] &]
    Select[Range[150],MemberQ[realDigitsRecip[#],8]&] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jan 11 2025 *)

A020764 Decimal expansion of 1/sqrt(7).

Original entry on oeis.org

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

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Comments

1/sqrt(7) = 0.377964473009227227214516536234180060815751311868921454338333494171... - Vladimir Joseph Stephan Orlovsky, May 27 2010

Crossrefs

Cf. A010465.

Programs

Formula

Equals 1/A010465.

A021337 Decimal expansion of 1/333.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3
Offset: 0

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Examples

			0.00300300300300300300300300300300300300300300300300...
		

Programs

Formula

G.f.: 3*x^2/(1 - x^3). - Chai Wah Wu, Jun 21 2016
From Wesley Ivan Hurt, Jul 02 2016: (Start)
a(n) = a(n-3) for n>2.
a(n) = 1 - cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3).
a(n) = 3*(1 - sgn((n+1) mod 3)).
a(n) = 1 + (n mod 3) - ((n+1) mod 3). (End)
Showing 1-10 of 74 results. Next