A177413 -id:A177413 - cp's OEIS frontend

A184952 High water marks in A177413.

0, 1, 2, 4, 40, 52, 82, 4839, 5813, 8366, 11153, 46254, 1040968, 12925493

Offset: 1

Ben Branman, Dec 21 2011

12925493 does not appear in A177413 until n=292558.

			The first few terms of the continued fraction of the Dottie number are 0, 1, 2, 1, 4, 1, 40 of which the high water marks are 0, 1, 2, 4, 40...

Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957, A177413, A185185.

z = x /. FindRoot[x == Cos[x], {x, 0},
   WorkingPrecision -> 100000]; data = ContinuedFraction[z];
g[list_] :=
Delete[list,
  Transpose[{DeleteCases[
     Table[If[list[[n]] < list[[n - 1]], n, no], {n, 2,
       Length[list]}], no]}]];

Offset changed by Andrew Howroyd, Aug 10 2024

A185185 Indices of record values in A177413.

Original entry on oeis.org

0, 1, 2, 4, 6, 19, 29, 53, 2035, 3995, 5328, 10141, 14675, 292557

Offset: 1

Ben Branman, Dec 22 2011

Eric Weisstein's World of Mathematics, Dottie Number.

Cf. A184952, A177413, A003957.

z = x /. FindRoot[x == Cos[x], {x, 0},
   WorkingPrecision -> 1000000]; data = ContinuedFraction[z];
g[list_] :=
Delete[list,
  Transpose[{DeleteCases[
     Table[If[list[[n]] < list[[n - 1]], n, no], {n, 2,
       Length[list]}], no]}]]; p =
FixedPoint[g, DeleteDuplicates[data]]; Flatten[
Map[Position[data, #, 1, 1] &, p, {1}]]-1

A177413(a(n)) = A184952(n).

Offset changed by Andrew Howroyd, Aug 10 2024

A003957 The Dottie number: decimal expansion of root of cos(x) = x.

Original entry on oeis.org

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

Offset: 0

Leonid Broukhis

Let P be the point in quadrant I where the curve y=sin(x) meets the circle x^2+y^2=1. Let d be the Dottie number. Then P=(d,sin(d)), and d is the slope at P of the sine curve. - Clark Kimberling, Oct 07 2011
From Ben Branman, Dec 28 2011: (Start)
The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator users before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.
The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian small letter ayb (ա, the first letter in the Armenian alphabet) to denote this constant. (End)

			0.73908513321516064165531208767387340401341175890075746496568063577328...

H. Arakelian, The Fundamental Dimensionless Values (Their Role and Importance for the Methodology of Science). [In Russian.] Yerevan, Armenia: Armenian National Academy of Sciences, 1981.
A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..499 from Ben Branman)
Hrant Arakelian, New Fundamental Mathematical Constant: History, Present State and Prospects, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.
Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Trefor Bazett, What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(...?? // Banach Fixed Point Theorem, YouTube video, 2022.
J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
Steven Finch, Exercises in Iterational Asymptotics II, arXiv preprint (2025). arXiv:2501.06065 [math.NT]
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
T. H. Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
David R. Stoutemyer, Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses, arXiv:2207.00707 [math.GM], 2022.
Eric Weisstein's World of Mathematics, Dottie Number
Index entries for transcendental numbers.

Cf. A009442, A177413, A182503, A197002, A200309, A212112, A212113, A217066.

Cf. A330119 (degrees-based analog).

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

RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]
FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]][[1]] (* Ben Branman, Apr 12 2008 *)
N[NestList[Cos, 1, 100], 20] (* Clark Kimberling, Jul 01 2019 *)
RealDigits[Root[{# - Cos[#] &, 0.739085}], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
RealDigits[Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)

solve(x=0,1,cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011

from sympy import Symbol, nsolve, cos
x = Symbol("x")
a = list(map(int, str(nsolve(cos(x)-x, 1, prec=110))[2:-2]))
print(a) # Michael S. Branicky, Jul 15 2022

Equals twice A197002. - Hugo Pfoertner, Feb 20 2024

More terms from David W. Wilson

Additional references from Ben Branman, Dec 28 2011

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

A208672 a(n) = floor[1/(1-f(n))], where f(n) is the least nonnegative number such that f(n)^n = cos(f(n)).

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 23, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 41, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 89, 90, 92

Offset: 0

Ben Branman, Feb 29 2012

nonn

For n=0, the only possible solution is f(0)=0, which yields a(0)=1.
f(n)->1 as n->infinity.
a(n) ~ -n/log(cos(1))
f(1) = the Dottie number 0.73908513321516 = A003957
f(2) is A125578
f(3) is A125579
a(n) is defined for negative values of n as well.
If we let a(n)=floor[c(n)], c(n)=1/(1-f(n)), then f(n)^n=cos(f(n)) <=> 1-1/c(n) = cos(1-1/c(n))^(1/n) = exp(log(cos(1-1/c(n)))/n) = exp(log(cos(1)+O(1/c(n)^2))/n) = 1+log(cos(1))/n+o(1/n), assuming c(n) ~ c*n, which then yields c = -1/log(cos(1)). - M. F. Hasler, Mar 05 2012

			For n=4, the only positive solution to x^4=cos(x) is x=0.890553, so a(4)=floor(1/(1-.890553)) = floor(9.13682) = 9, so a(4)=9.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dottie Number
Eric Weisstein's World of Mathematics, Cosine
Eric Weisstein's World of Mathematics, Fixed Point

Cf. A003957, A125578, A125579, A177413.

f[n_] := 1/(1 - FindRoot[x^n == Cos[x], {x, 0, 1}, WorkingPrecision -> 1000][[1,2]]); Table[Floor[f[n]], {n, 0, 100}]

a(n)=1\(1-solve(x=0,1,x^n-cos(x))) \\ Charles R Greathouse IV, Mar 04 2012

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A184952 High water marks in A177413.

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A185185 Indices of record values in A177413.

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A003957 The Dottie number: decimal expansion of root of cos(x) = x.

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

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

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A208672 a(n) = floor[1/(1-f(n))], where f(n) is the least nonnegative number such that f(n)^n = cos(f(n)).

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