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.

A197020 Decimal expansion of the radius of the circle tangent to the curve y=cos(2x) at points (x,y) and (-x,y), where 0

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 08 2011

Keywords

Comments

Let (x,y) denote the point of tangency. Then
x=0.371462711321448956555130330602759317162856415...
y=0.736492824477900896049098419167188850255855384...
slope=-1.3528907117613955482765053348775509428929...
(The Mathematica program includes a graph.)

Examples

			0.461923187705228238217153033369389999620434726705688657976706430379511394115251....

Crossrefs

Cf. A197021, A196022, A196026, A197027, A197016.

Programs

  • Mathematica
    r = .462; c = 2;
Show[Plot[Cos[c*x], {x, -2, 2}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
t = x /. FindRoot[c*Sin[c*x] Cos[c*x] - x ==
    x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
RealDigits[t] (* x coordinate of tangency point *)
y = Cos[c*t]  (* y coordinate of tangency point *)
radius = Cos[c*t] - t/(c*Sin[c*t]) (* A197020 *)
RealDigits[radius]
slope = -c*Sin[c*t] (* slope at tangency point *)
  • PARI
    t=solve(x=.3,.4, 2*sin(2*x)*cos(2*x) - x*sqrt(1 + 4*sin(2*x)^2) - x); cos(2*t) - t/(2*sin(2*t)) \\ Charles R Greathouse IV, Feb 04 2025

A195947 E.g.f. satisfies: A(x) = Sum_{n>=0} (-1)^n/n! * Sum_{k=0..n} (-1)^k*C(n,k)*(1 + x*A(x)^k)^k.

Original entry on oeis.org

1, 1, 5, 58, 1093, 28731, 971719, 40236449, 1972617385, 111779567596, 7189852342091, 517600784497237, 41237095369088029, 3602389000897583001, 342422738142493542031, 35186740743134660359186, 3887047020291801938191057, 459397561144034558519708403
Offset: 0

Views

Author

Paul D. Hanna, Sep 27 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1093*x^4/4! + 28731*x^5/5! +...
where:
A(x) = 1 + A(x)*exp(A(x) - 1)*x + A(x)^4*exp(A(x)^2 - 1)*x^2/2! + A(x)^9*exp(A(x)^3 - 1)*x^3/3! + A(x)^16*exp(A(x)^4 - 1)*x^4/4! +...
Also, e.g.f. A = A(x) satisfies:
A(x) = 1 - (1 - (1+x*A)) + 1/2!*(1 - 2*(1+x*A) + (1+x*A^2)^2) -
1/3!*(1 - 3*(1+x*A) + 3*(1+x*A^2)^2 - (1+x*A^3)^3) +
1/4!*(1 - 4*(1+x*A) + 6*(1+x*A^2)^2 - 4*(1+x*A^3)^3 + (1+x*A^4)^4) -
1/5!*(1 - 5*(1+x*A) + 10*(1+x*A^2)^2 - 10*(1+x*A^3)^3 + 5*(1+x*A^4)^4 - (1+x*A^5)^5) +-...

Crossrefs

Cf. A196022, A195895.

Programs

  • PARI
    {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1,n,exp(A^m-1)*A^(m^2)*X^m/m!)); n!*polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, 1/m!*sum(k=0, m, binomial(m, k)*(-1)^(m-k)*(1+X*A^k)^k))); n!*polcoeff(A, n)}

Formula

E.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^(n^2)*exp(A(x)^n - 1)*x^n/n!.

A197021 Decimal expansion of the radius of the circle tangent to the curve y=cos(3x) at points (x,y) and (-x,y), where 0

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 08 2011

Keywords

Comments

Let (x,y) denote the point of tangency. Then
x=0.346818914654599529577486037538498433565584415464...
y=0.505826306518745297430716717373078359704411629139...
slope=-2.5879060509806840663013781941932136174746999...
(The Mathematica program includes a graph.)

Examples

			radius=0.3718110417361721840195647351588579028970626...

Crossrefs

Cf. A197020, A196022, A196026, A197027, A197016.

Programs

  • Mathematica
    r = .371; c = 3;
Show[Plot[Cos[c*x], {x, -0.5, 0.5}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
t = x /. FindRoot[c*Sin[c*x] Cos[c*x] - x == x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
RealDigits[t]   (* x coordinate of tangency point *)
y = Cos[c*t]    (* y coordinate of tangency point *)
radius = Cos[c*t] - t/(c*Sin[c*t])
RealDigits[radius]   (* A197021 *)
slope = -c*Sin[c*t]  (* slope at tangency point *)

A197029 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(4x) at points (x,y), (-x,y).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

Let (x,y) denote the point of tangency, where x>0:
x=0.488618197079923270050681129865078039260837...
y=0.374332154777652501331094642913853652491893...
slope=3.709178750935618333987343550424591912283...
(The Mathematica program includes a graph.)

Examples

			radius=0.5060643332165245100546376217734714411...

Crossrefs

Cf. A197026, A196027, A196028, A196022.

Programs

  • Mathematica
    r = .5; c = 4;
Show[Plot[-Cos[c*x], {x, -1, 1}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1.5, 2}], PlotRange -> All, AspectRatio -> Automatic]
u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, .4, .5}, WorkingPrecision -> 100]
t = Re[t1]    (* x coordinate of tangency point *)
y = -Cos[c*t] (* y coordinate of tangency point *)
radius = u[t]
RealDigits[radius] (* A197029 *)
slope = c*Sin[c*t] (* slope at tangency point *)
Showing 1-4 of 4 results.

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