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.

Previous Showing 11-17 of 17 results.

A102015 Continued fraction expansion of smallest positive root of tan(x) = x.

Original entry on oeis.org

4, 2, 37, 2, 3, 4, 5, 1, 2, 1, 1, 2, 1, 5, 3, 3, 3, 5, 52, 1, 40, 2, 2, 20, 3, 2, 3, 12, 1, 19, 18, 1, 1, 24, 1, 8, 3, 2, 1, 2, 2, 4, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 2, 17, 4, 3, 1, 2, 2, 3, 1, 7, 1, 1, 6, 31, 13, 13, 3, 5, 2, 2, 1, 1, 1, 1, 1, 27, 2, 2, 9, 1, 6, 1, 1, 1, 2, 3, 2, 2, 1, 3, 1, 4, 3, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jun 19 2007

Keywords

Crossrefs

Cf. A115365 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[BesselJZero[3/2, 1], 100] (* Amiram Eldar, Jun 09 2021 *)

Extensions

Offset changed by Andrew Howroyd, Aug 04 2024

A255272 Decimal expansion of the second smallest positive root of tan(x) = x.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Feb 20 2015

Keywords

Comments

This constant is quite close to 5*Pi/2 - 1/8 = 7.72898...
Searching for solutions x=k*Pi+Pi/2-e and small e, for k=1,2,3.... means via the approximation tan(x) = 1/e-e/3-e^3/45... that e is approximately 1/(k*Pi+Pi/2), so the constants x are close to k*Pi+Pi/2-1/(k*Pi+Pi/2). Here k=2 and the constant is close to 5*Pi/2-2/(5*Pi) = 7.7266576... - R. J. Mathar, Jul 11 2024

Examples

			7.72525183693770716419506893306298662637815930461...

Links

Crossrefs

Cf. A115365 (smallest positive root), A062546 (C_2 = 2nd du Bois-Reymond constant), A224196 (C_3), A207528 (C_4), A243108 (C_5), A245333 (C_6).
Cf. A079330, A088989.

Programs

  • Mathematica
    xi[n_] := x /. FindRoot[Tan[x] == x, {x, n*Pi + Pi/2 - 1/(4*n)}, WorkingPrecision -> 102]; RealDigits[xi[2]] // First
  • PARI
    solve(x=7,7.8,tan(x)-x) \\ Charles R Greathouse IV, Apr 20 2016

A328227 Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.

Original entry on oeis.org

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

Views

Author

Jack Zhang, Oct 08 2019

Keywords

Comments

We are in a rowboat on a circular lake, starting at the center. At the edge of the lake is a mean goblin. He can run k times as fast as we can row. This is the minimum value of k such that we will not be able to escape.
From Rian Hunter, Jun 16 2021: (Start)
For a spirograph defined by complex function z = p * e^(-i * b * t) + b * e^(i * t), this is the value of p as b->oo such that each petal is tangent to the next one.
If we consider the set of all right triangles such that their tangent value is equal to the opposite angle in radians, this value is equal to the negative secant of the right triangle from that set with the smallest nonzero opposite angle. (End)
The envelope of the t*x = sin(t*y) family of curves contains the set of y = (-1)^n*k_n*x straight lines (n > 0), where k_n is the solution of (n*Pi + arccos(1/k))^2 + 1 = k^2. This entry is k_1. See illustration, section Links. - Luc Rousseau, Mar 11 2022
Maximum negative value of x/sin(x). - Andrew Slattery, Jun 29 2022

Examples

			4.6033388487517003525565820291030165130673971341605323460394301154384587319659...

Links

Crossrefs

Cf. A115365.
Equals 1/A213053.

Programs

  • Mathematica
    NSolve[x^2==1+(Pi+ArcCos[1/x])^2,x,Reals,WorkingPrecision->100]
  • PARI
    solve(x=4, 5, 1 + (Pi+acos(1/x))^2 - x^2) \\ Michel Marcus, Oct 08 2019

Formula

x=-sec(y), where decimal expansion of y is A115365.
Alternatively, x=sqrt(y^2+1).

A362219 Decimal expansion of smallest positive solution to tan(x) = arctan(x).

Original entry on oeis.org

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

Views

Author

Wolfe Padawer, Apr 11 2023

Keywords

Examples

			4.067588865765862790917085025312411319068300674493957922637263436551...

Crossrefs

Cf. A115365, A362220.

Programs

  • Mathematica
    RealDigits[FindRoot[Tan[x] == ArcTan[x], {x, 4}, WorkingPrecision -> 105][[1, 2]]][[1]]
  • PARI
    solve(x=4, 4.5, tan(x)-atan(x)) \\ Michel Marcus, Apr 12 2023

Formula

Equals tan(A362220).

A362220 Decimal expansion of smallest positive root of x = tan(tan(x)).

Original entry on oeis.org

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

Views

Author

Wolfe Padawer, Apr 11 2023

Keywords

Examples

			1.329731220678945515737146065584648589604829857490380436751246457979...

Crossrefs

Cf. A115365, A362219.

Programs

  • Mathematica
    RealDigits[FindRoot[Tan[Tan[x]] == x, {x, 1.3}, WorkingPrecision -> 105][[1, 2]]][[1]]
  • PARI
    solve(x=1.32, 1.35, tan(tan(x)) - x) \\ Michel Marcus, Apr 12 2023

Formula

Equals tan(A362219).
Equals arctan(A362219).

A242055 Decimal expansion of c, a constant appearing in the asymptotic lower bound of the size of a restricted difference set.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Aug 13 2014

Keywords

Examples

			1.560277942041879702102077381568463756373995745949542538537...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.28 p. 188.

Links

Crossrefs

Cf. A115365.

Programs

  • Mathematica
    digits = 103; theta = t /. FindRoot[Tan[t] == t, {t, 4}, WorkingPrecision -> digits+5]; c = Sqrt[2*(1 - Sin[theta]/theta)]; RealDigits[c, 10, digits] // First

Formula

c = sqrt(2*(1 - sin(theta)/theta)), where theta is the smallest positive zero of tan(t)-t (theta = A115365).

A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).

Original entry on oeis.org

1, 4, 0, 8, 5, 9
Offset: 0

Views

Author

Bernard Schott, Apr 23 2021

Keywords

Comments

The equation of the sinc function is y = sin(x)/x.
Equivalently, sum of f(x) = sinc(x) where x > 0 and f'(x) = 0. - David A. Corneth, May 01 2021
These extreme values are obtained when x_k > 0 is a solution to tan(x) = x (see Chronomath link), or equivalently to y = tanc(x) = tan(x)/x = 1. The corresponding k-th extreme value is y_k = sin(x_k)/x_k.
Every extremum y_k = (-1)^k/(k*Pi) + O(1/k^2), hence the series Sum_{k > 0} sin(x_k)/x_k is convergent.
However, this series is not absolutely convergent, just as (C_1)/2 diverges where C_1 is the corresponding du Bois-Reymond constant.

Examples

			-0.140859...

References

  • Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.18, pp. 285 and 303.

Links

Crossrefs

Coordinates of the 1st extremum: A115365 (x_1), A213053 (y_1).
Cf. A062546, A224196, A207528, A243108, A245333.

Formula

Equals Sum_{k >= 1} sinc(x_k) or Sum_{k >= 1} (-1)^k / sqrt(1+(x_k)^2), where x_k is the k-th positive root of x = tan(x).

Extensions

More terms from Amiram Eldar, Apr 23 2021
Name clarified by N. J. A. Sloane, May 01 2021
Previous Showing 11-17 of 17 results.

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

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