A003957 -id:A003957 - cp's OEIS frontend

A197002 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/4) orthogonally.

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

Offset: 0

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=0.36954256660758032082765604383693...
yo=0.40397275329951720931896174006631...
m=1.093169744985016922088153214160579...
|OP|=0.547499492185436214325204150357...

Index entries for transcendental numbers.

Cf. A197003, A196996, A197000, A003957.

evalf(solve(cos(x)=x,x)/2, 140);  # Alois P. Heinz, Feb 20 2024

c = Pi/4;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197002 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197003 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

solve(x=0,1,cos(x)-x)/2  \\ Gleb Koloskov, Jun 16 2021

Equals d/2 = A003957/2, where d is the Dottie number. - Gleb Koloskov, Jun 16 2021

A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 06 2003

This constant can also be defined as the root of coth x = x, as this equation and the above are equivalent. - Carl R. White, Dec 09 2003. Also the root of x*tanh x = 1. - N. J. A. Sloane, May 07 2020
This constant is also the point on the parametric tractrix (t - tanh(t), sech(t)) the least distant from the origin. - Michael Clausen, Feb 18 2013
This constant also equals sqrt(lambda^2+1), where lambda is the Laplace limit constant A033259. - Jean-François Alcover, Sep 08 2014, after Steven Finch.
For each of the real symmetric n X n matrices M defined by M(i,j) = max(i,j) with n >= 2, there exist n-1 negative eigenvalues < -1/4 and only one positive eigenvalue lambda(n) such that n^2/2 < lambda(n) < n^2. Indeed, when n tends to infinity, lambda(n) ~ n^2/(this constant)^2 (see reference O. Carton et al.). For n = 2, the positive eigenvalue is (3+sqrt(17))/2 [A178255]. - Bernard Schott, Mar 13 2020

			1.1996786402577338339163698486411419442614587884186072...

O. Carton, L. Rosaz, M. Zeitoun, Problèmes corrigés de Mathématiques posés au Concours de Mines/Ponts, Tome 5, Ellipses, 1992; Problème Mines-Ponts 1991 - Options M, P', TA - Epreuve pratique p. 125.
Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jogundas Armaitis, Molecules and Polarons in Extremely Imbalanced Fermi Mixtures, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.
Robert Ferréol, Tractrix, Mathcurve.
D. E. Knuth, Whirlpool Permutations, May 05 2020.
Mathematics Stack Exchange, Laplace limit constant.
Eric Weisstein's World of Mathematics, Kepler's Equation.
Eric Weisstein's World of Mathematics, Laplace Limit.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Wikipedia, Tractrix.
Index entries for transcendental numbers.

Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.

RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)
1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //
  (* Aaron Hendrickson, Mar 17 2021 *)

solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */

Equals 1 + 2*Sum_{n>=1} (Laguerre(n-1,1,4n)/n)*e^(-2n) (see Mathematics Stack Exchange in Links). - Aaron Hendrickson, Mar 17 2022

A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2020

Let S and C denote the graphs of y = sin x and y = cos x. For each point (u, sin u) on S, let S(u) be the line normal to S at (u, sin u), and let (snc u, cos(snc u)) be the point of intersection of S(u) and C. Let d(u) be the distance from (u,sin u) to (snc u, cos(snc u)). We call d(u) the u-normal distance from S to C and note that in [0,Pi], there is a unique number u' such that d(u') > d(u) for all real numbers u except those of the form u' + k*Pi. We call d(u') the maximal normal distance between sine and cosine, and we call snc the sine-normal-to-cosine function.
For each (v, cos v) on C, let C(u) be the line normal to C at (v, cos v), and let (cns v, sin(cns v)) be the point of intersection of C(u) and S. Let e(v) be the distance from (v, cos v) to (cns v, sin(cns v)). We call d(v) the v-normal distance from C to S and note that there exists a unique number v' that maximizes e, and e(v') = d(u'). We call cns the cosine-normal-to-sine function. The numbers u' and v' are given in A332501 and A332503.
Note that the maximal normal distance (see Example) exceeds the normal distance from (Pi/2,1) in sine to (Pi/2,0) in cosine - possibly a surprise!

			2.72573705679992524967463858129656... = the number u in [0,2 Pi] such that the line normal to S at (u, sin u) passes through the point (3 Pi/4,0); cf. A332501.
0.4039727532995172093189617400663... = sin u; cf. A086751.
1.0949989843708724286504083007155... = maximal normal distance between sine and cosine.
1.9866519235847646080193264936226... = snc u; cf A332503.

Index entries for transcendental numbers.

Cf. A086751, A332501, A332502, A332503, A332504, A332505, A332506, A332507, A003957.

Plot[{Sin[x], Cos[x]}, {x, -Pi, 3 Pi}, AspectRatio -> Automatic,
ImageSize -> 600, PlotLabel -> "sine and cosine"]
t = Table[x = x /. FindRoot[Cos[x] == -x Sec[u] + u Sec[u] + Sin[u], {x, 0}], {u, -2 Pi, 2 Pi, Pi/101}];
ListPlot[t, PlotLabel -> "y \[Equal] snc(x)"]
ListPlot[Cos[t], PlotLabel -> "y \[Equal] cos(snc(x))"]
t = Table[x = x /. FindRoot[Sin[x] == x Csc[u] - u Csc[u] + Cos[u], {x, 0.1}], {u, -2 Pi + .01, 2 Pi - .01, Pi/101}];
ListPlot[t, PlotLabel -> "y \[Equal] cns(x)"]
ListPlot[Sin[t], PlotLabel -> "y \[Equal] sin(cns(x))"]
u = u /. FindRoot[0 == (-3 Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 1}, WorkingPrecision ->120]  (* A332501 *)
y = Sin[u]  (* A086751 *)
d = 2*Sqrt[(u - 3 Pi/4)^2 + y^2]  (* A332500 *)
RealDigits[u][[1]]  (* A332501 *)
RealDigits[y][[1]]  (* A086751 *)
RealDigits[d][[1]]  (* A332500 *)

my(d=solve(x=0,1,cos(x)-x)); sqrt(d^2+2-2*sqrt(1-d^2)) \\ Gleb Koloskov, Jun 16 2021

d(u') = 2*sqrt((u - 3 Pi/4)^2 + (sin u)^2).

Equals sqrt(d^2+2-2*sqrt(1-d^2)) where d = A003957. - Gleb Koloskov, Jun 16 2021

A177413 Continued fraction expansion of the constant x defined by x = cos(x).

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 40, 1, 9, 4, 2, 1, 15, 2, 12, 1, 21, 1, 17, 52, 3, 1, 6, 1, 2, 2, 1, 1, 2, 82, 1, 1, 2, 39, 27, 1, 15, 1, 1, 1, 2, 26, 1, 10, 1, 2, 1, 1, 1, 6, 1, 4, 1, 4839, 1, 2, 1, 4, 7, 2, 1, 43, 1, 21, 1, 5, 2, 1, 6, 4, 9, 2, 19, 1, 1, 2, 1, 1, 2, 38

Offset: 0

Ben Branman, Dec 10 2010

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957.

FindRoot[x == Cos[x], {x, 0}, WorkingPrecision -> 10000]; z = x /. %; ContinuedFraction[z]

A086751 Decimal expansion of the solution to x*sqrt(1-x^2) + arcsin(x) = Pi/4, or the length of the line connecting the origin to the center of the chord of a circle, centered at 0 and of radius 1, that divides the circle such that 1/4 of the area is on one side and 3/4 is on the other side.

Original entry on oeis.org

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

Offset: 0

Jonathan R. Anderson (neo__jon(AT)hotmail.com), Jul 30 2003

Decimal expansion of the number sin(u'), where u' is the number in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0). See A332500. - Clark Kimberling, May 05 2020

			0.403972753299517...

Robert P. P. McKone, Table of n, a(n) for n = 0..19999

Cf. A003957.

Digits := 240 ; x := 0.4 ; for i from 1 to 8 do f := sin(2.0*x)+2.0*x-Pi/2.0 ; fp := 2*cos(2*x)+2.0 ; x := x-evalf(f/fp) ; printf("%.120f\n",sin(x)) ; od: x := sin(x) ; read("transforms3") ; CONSTTOLIST(x) ; # R. J. Mathar, May 19 2009

digits = 105; Sin[FindRoot[Sin[2*a]/2+a == Pi/4, {a, 1/2}, WorkingPrecision -> digits][[1, 2]]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)

solve(x=0, 1, x*sqrt(1-x^2) + asin(x) - Pi/4) \\ Michel Marcus, May 05 2020

my(d=solve(x=0,1,cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/2 \\ Gleb Koloskov, Jun 16 2021

Define k(n+1) as k(n) - (k(n)*sqrt(1-k(n)^2) + arcsin(k(n)) - Pi/4). The sequence is the decimal expansion of lim_{n -> infinity} k(n).

Equals sqrt(2-2*sqrt(1-d^2))/2, where d = A003957 is the Dottie number. - Gleb Koloskov, Jun 16 2021

More terms from Jim Nastos, Sep 05 2003

More digits from R. J. Mathar, May 19 2009

A124597 Decimal expansion of the solution to x^2 = sin(x).

Original entry on oeis.org

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

Offset: 0

Denton J. Dailey (djd1497(AT)aol.com), Dec 18 2006

Fixed point of cardinal sine (sinc) function. - Michal Paulovic, Jun 13 2023

			0.87672621539506244597211864314192813997168545140130499558964277342...

Index entries for transcendental numbers.

Cf. A003957.

Digits:=105;fsolve(sin(x)/x-x,x); # Michal Paulovic, Jun 13 2023

RealDigits[ FindRoot[ Sin[x] == x^2, {x, {.7, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]

solve(x=0.8,0.9,sin(x)/x-x) \\ Michal Paulovic, Jun 13 2023

More terms from Robert G. Wilson v, Dec 21 2006

A200309 Expansion of e.g.f.: 1/(cos(x) - x).

Original entry on oeis.org

1, 1, 3, 12, 65, 440, 3571, 33824, 366113, 4458240, 60321091, 897774592, 14576528801, 256391130112, 4856647308787, 98567413125120, 2133825372539585, 49080991762153472, 1195339768057071619, 30729146849826701312, 831545527540481198465, 23627123985544955559936

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

Radius of convergence |x| < r, where r = cos(r) = 0.739085133215160... (A003957).

The continued fraction converges in the whole complex plane, cut along |z|=infinity.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 65*x^4/4! + 440*x^5/5! +...
where 1/A(x) = 1 - x - x^2/2! + x^4/4! - x^6/6! +...
Special values.
A(Pi/5) = 10/(5*(sqrt(5)+1)/2 - 2*Pi) = 5.534081362740...
A(Pi/6) = 6/(3*sqrt(3) - Pi) = 2.920333635550...
A(Pi/8) = 8/(4*sqrt(2+sqrt(2)) - Pi) = 1.882599403781...
A(Pi/10) = 10/(5*sqrt(10+sqrt(20))/2 - Pi) = 1.5701119741529...
A(Pi/12) = 12/(6*sqrt(2+sqrt(3)) - Pi) = 1.4201994774470...

G. C. Greubel, Table of n, a(n) for n = 0..427
S. N. Gladkovskii, Analysis Of The Continued Fractions (in Russian).

Cf. A006154, A003957.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(Cos(x) - x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 10 2018

CoefficientList[Series[1/(Cos[x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

{a(n)=n!*polcoeff(1/(cos(x+x*O(x^n))-x),n)}

x='x+O('x^30); Vec(serlaplace(1/(cos(x) - x))) \\ G. C. Greubel, Jul 10 2018

E.g.f.: E(x)=1/(cos(x) - x) = (1-x^2/((x-1)*G(0) + x^3))/(1-x); G(k)= 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 08 2012
E.g.f.: 1/(G(0) - x) where G(k) =  1 - x^2/((4*k+1)*(4*k+2) - x^2*(4*k+1)*(4*k+2)/(x^2 - 4*(k+1)*(4*k+3)/G(k+1) )); - Sergei N. Gladkovskii, Dec 16 2012
a(n) ~ n!/((sin(r)+1)*r^(n+1)), where r = 0.73908513321516... is the root of the equation r = cos(r). - Vaclav Kotesovec, Jun 27 2013

A302977 Numerators of the rational factor of Kaplan's series for the Dottie number.

Original entry on oeis.org

1, -1, -1, -43, -223, -60623, -764783, -107351407, -2499928867, -596767688063, -22200786516383, -64470807442488761, -3504534741776035061, -3597207408242668198973, -268918457620309807441853, -185388032403184965693274807, -18241991360742724891839902347

Offset: 0

Ozaner Hansha, Apr 16 2018

In Kaplan's original article, where the term "Dottie" was coined, he mentioned that while the number was indeed transcendental, it was possible to express it as an infinite sum with the general term r_n*Pi^(2n+1) where r_n was a sequence of rational numbers.

			The partial Kaplan series at n=3 is d = Pi/4 - Pi^3/768 - Pi^5/61440 - 43*Pi^7/165150720.

Amiram Eldar, Table of n, a(n) for n = 0..100
J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
Ozaner Hansha, The Dottie Number.
Ozaner Hansha, Kaplan's series
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.

Cf. A003957, A177413, A182503, A200309, A212112, A212113.

Cf. A306254 (denominators).

f[x_] := x - Cos[x]; g[x_] := InverseFunction[f][x]; s = {1}; Do[AppendTo[s, Numerator[(-1/2)^n * 1/n! * Derivative[n][g][Pi/2]]], {n, 3, 30, 2}]; s (* Amiram Eldar, Jan 31 2019 *)

These are the numerators of the unique sequence of rational numbers r_n such that d = Sum_{n>=0} r_n*Pi^(2*n+1) (where d is the Dottie number A003957).

r_0 = 1/4 and for n>0, r_n = b_(2*n+1); where b_n = g^(n)(Pi/2)/(2^n*n!) (and g^(n) is the n-th derivative of the inverse of x - cos(x)). A proof of this can be found in the second Hansha link.

More terms from Amiram Eldar, Jan 31 2019

A306254 Denominators of the rational factor of Kaplan's series for the Dottie number.

Original entry on oeis.org

4, 768, 61440, 165150720, 47563407360, 669692775628800, 417888291992371200, 2808209322188734464000, 3055331742541343096832000, 33437550590372458851729408000, 56175084991825730870905405440000, 7276695809501137874093602599075840000, 17464069942802730897824646237782016000000

Offset: 0

Amiram Eldar, Feb 01 2019

These are the denominators of the unique sequence of rational numbers r_n such that d = Sum_{n>=0} r_n*Pi^(2*n+1) (where d is the Dottie number A003957). The numerators are in A302977.

			The Kaplan series begins with d = Pi/4 - Pi^3/768 - Pi^5/61440 - 43*Pi^7/165150720 - ...

Amiram Eldar, Table of n, a(n) for n = 0..100
Ozaner Hansha, The Dottie Number.
Ozaner Hansha, Kaplan's series.
Samuel R. Kaplan, The Dottie Number, Mathematics Magazine, Vol. 80, No. 1 (2007), pp. 73-74, alternative link.

Cf. A003957, A177413, A182503, A200309, A212112, A212113, A302977.

f[x_] := x - Cos[x]; g[x_] := InverseFunction[f][x]; s = {}; Do[AppendTo[s, Denominator[(-1/2)^n * 1/n! * Derivative[n][g][Pi/2]]], {n, 1, 30, 2}]; s

A125578 Decimal expansion of positive root of x^2 = cos(x).

Original entry on oeis.org

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

Offset: 0

Denton J. Dailey (djd1497(AT)aol.com), Jan 03 2007

Both roots have same magnitude: x = +-0.824132312302...

Fixed point of cos(x)/x function. - Michal Paulovic, Jun 13 2023

			0.824132312302522422960956785771991108142689866748289991732...

Cf. A003957, A124597.

Digits:=100;fsolve(cos(x)/x-x,x); # Michal Paulovic, Jun 13 2023

RealDigits[FindRoot[Cos[x] == x^2, {x, {.7, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]

solve(x=0,1,cos(x)-x^2) \\ Charles R Greathouse IV, Apr 14 2014

cp's OEIS Frontend

A197002 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/4) orthogonally.

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A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

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A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.

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A177413 Continued fraction expansion of the constant x defined by x = cos(x).

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A086751 Decimal expansion of the solution to x*sqrt(1-x^2) + arcsin(x) = Pi/4, or the length of the line connecting the origin to the center of the chord of a circle, centered at 0 and of radius 1, that divides the circle such that 1/4 of the area is on one side and 3/4 is on the other side.

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A124597 Decimal expansion of the solution to x^2 = sin(x).

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A200309 Expansion of e.g.f.: 1/(cos(x) - x).

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