A003957 -id:A003957 - cp's OEIS frontend

A212112 Numerators of convergents to the Dottie number, A003957.

0, 1, 2, 3, 14, 17, 694, 711, 7093, 29083, 65259, 94342, 1480389, 3055120, 38141829, 41196949, 903277758, 944474707, 16959347777, 882830559111, 2665451025110, 3548281584221, 23955140530436

Offset: 0

Ben Branman, May 01 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957, A212113.

z = FindRoot[Cos[x] == x, {x, 0, 1}, WorkingPrecision -> 10000][[1, -1]]; Numerator[Convergents[z, 100]]

A212113 Denominators of convergents to the Dottie number, A003957.

Original entry on oeis.org

1, 1, 3, 4, 19, 23, 939, 962, 9597, 39350, 88297, 127647, 2003002, 4133651, 51606814, 55740465, 1222156579, 1277897044, 22946406327, 1194491026048, 3606419484471, 4800910510519, 32411882547585

Offset: 0

Ben Branman, May 01 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957, A212112.

z = FindRoot[Cos[x] == x, {x, 0, 1}, WorkingPrecision -> 10000][[1, -1]]; Denominator[Convergents[z, 100]]

A182503 Engel expansion of the Dottie number, A003957.

Original entry on oeis.org

2, 3, 3, 4, 5, 15, 17, 66, 196, 233, 284, 375, 1613, 2131, 3574, 14122, 24171, 49097, 56871, 69361, 193406, 243145, 289951, 682749, 14501588, 21191773, 121635191, 810759781, 1292785381, 136110231377, 294401497761

Offset: 1

Ben Branman, May 02 2012

Dottie number = 1/2 + 1/2/3 + 1/2/3/3 + 1/2/3/3/4 + 1/2/3/3/4/5 + 1/2/3/3/4/5/15 +...

G. C. Greubel, Table of n, a(n) for n = 1..500
Jean-Christophe Pain, An exact series expansion for the Dottie number, arXiv:2303.17962 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Dottie Number
Index entries for sequences related to Engel expansions

Cf. A003957, A006784.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; z = FindRoot[x == Cos[x], {x, 1}, WorkingPrecision -> 10000][[1, -1]]; EngelExp[z, 30]

A330119 Degrees-based analog to the Dottie number (A003957).

Original entry on oeis.org

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

Offset: 0

Jon E. Schoenfield, Dec 01 2019

Using a calculator that has a cosine button and is set to calculate the values of trigonometric functions with the angles measured in degrees, start with any number and repeatedly hit the cosine button; the result will always converge to this value.

(If the calculator is set to calculate the values of trigonometric functions with the angles measured in radians rather than degrees, repeatedly hitting the cosine button will result in the value given at A003957.)

			0.9998477415310881129...

Cf. A003957.

solve(x=0, 1, cos(Pi*x/180)-x) \\ Michel Marcus, Dec 02 2019

A182101 Random walk determined by the binary digits of the Dottie number, A003957.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 7, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 7, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10

Offset: 0

Ben Branman, Apr 11 2012

Start at a(0)=0. Each 0 in the binary expansion corresponds to a step of -1, while a 1 corresponds to a step of +1.
Partial sums of the sequence 2*A121967(n)-1.
The first time a(n) is negative is n=93.

			a(5)=3, and the sixth bit of the Dottie number is 1, so a(6)=4.
On the other hand, the seventh bit of the Dottie number is 0, so a(7)=3.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A003957, A121967, A166006 (analogous sequence for Pi).

Accumulate[RealDigits[FindRoot[Cos[x] == x, {x, 0}, WorkingPrecision -> 1000][[1, -1]], 2][[1]] 2 - 1]

A100547 Decimal expansion of -cos(2^19*r) where r ~ 0.739085 is the root of cos x = x (A003957).

Original entry on oeis.org

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

Offset: 0

Gerald McGarvey, Jan 01 2005

			.999993333703844444728686845788780609012808601649593822014377814666899997...

Cf. A003957.

r = FindRoot[Cos[x] == x, {x, 1.}, WorkingPrecision -> 1000][[1, 2]];
RealDigits[Cos[2^19 r]][[1]] (* Ben Branman, Aug 19 2012 *)

A180619 Consider the function f(n)=1/(Abs(n-r)), where r is the Dottie number, A003957. Let g(n) be defined by the recursion g(n)=Cos(g(n-1)),g(0)=1. Now, a(n)=floor(f(g(n))).

Original entry on oeis.org

3, 5, 8, 11, 18, 26, 40, 58, 88, 130, 194, 287, 427, 633, 941, 1396, 2074, 3078, 4571, 6785, 10073, 14954, 22200, 32957, 48926, 72632, 107826, 160071, 237631, 352771, 523702, 777453, 1154157, 1713385, 2543579, 3776029, 5605645, 8321770, 12353952

Offset: 0

Ben Branman, Sep 12 2010

nonn

This sequence gives a sense of the rate of convergence to the Dottie Number.
Because higher values of a(n) means that g(n) is converging to the Dottie number, quick convergence means a high rate of increase for a(n).
This can be compared to other methods for approximation the Dottie number, by defining an analogous sequence.
This gives us an algorithm to measure the rate of convergence, for ANY function that convergence to a constant.
a(n) is asymptotically approaches an exponential regression.

			For n=3, g(3)=cos(cos(cos(1)))
f(g(3))~=11.7931005 So a(3)=floor(11.7931005)=11.

A369186 The denominators of a series that converges to the Dottie Number (A003957).

Original entry on oeis.org

1, 3, 12, 260, 5720, 314248, 17255072, 1769058016, 181357735680, 29880655637760, 4923158441956352, 1189676108826729472, 287484053261423565824, 95784714773484796761088, 31913810779214031287095296, 2804341960426298188743438336, 1232120770958699233546743119872

Offset: 1

Raul Prisacariu, Jan 15 2024

nonn

Whittaker's root series formula is applied to 1 - x - x^2/2! + x^4/4! - x^6/6! + ..., which is the Taylor expansion of cos(x) - x. The following infinite series for the Dottie number (D) is obtained: D = 1/1 - 1/3 + 1/12 - 3/260 + 1/5720 + 205/314248 - 4439/17255072 ... . The sequence is formed by the denominators of the series.

			a(1) is the denominator of -1/-1 = 1/1.
a(2) is the denominator of simplified -(-1/2!)/(-1* det ToeplitzMatrix((-1,1),(-1,-1/2!))) = (1/2)/(-3/2) = -1/3.
a(3) is the denominator of the simplified -det ToeplitzMatrix((-1/2!,-1),(-1/2!,0))/(det ToeplitzMatrix((-1,1),(-1,-1/2!))*det ToeplitzMatrix((-1,1,0),(-1,-1/2!,0))) = -(1/4)/((3/2)*-2) = 1/12.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A003957.

a(1)=1;

for n > 1, a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=-1, c(2)=-1/2!, c(3)=0, c(4)=1/4!, c(5)=0, c(6)=-1/6!, and c(n) is the coefficient of x^n in the Taylor expansion of cos(x)-x.

a(8)-a(17) from Chai Wah Wu, Feb 10 2024

A369187 The numerators of a series that converges to the Dottie Number (A003957).

Original entry on oeis.org

1, -1, 1, -3, 1, 205, -4439, 111021, -1724351, 2074717, 2567577481, -246042951203, 14444487376705, -726562139423955, 1473171168838825, 1178164765176836393, -204468301714665778099, 138848947223110087743421, -11701779801284441802592247, 7774256876827576332115737

Offset: 1

Raul Prisacariu, Jan 15 2024

sign

Whittaker's root series formula is applied to 1 - x - x^2/2! + x^4/4! - x^6/6! + ..., which is the Taylor expansion of cos(x) - x. The following infinite series for the Dottie number (D) is obtained: D = 1/1 - 1/3 + 1/12 - 3/260 + 1/5720 + 205/314248 - 4439/17255072 ... . The sequence is formed by the numerators of the series.

			a(1) is the numerator of -1/-1 = 1/1.
a(2) is the numerator of simplified -(-1/2!)/(-1* det ToeplitzMatrix((-1,1),(-1,-1/2!))) = (1/2)/(-3/2) = -1/3.
a(3) is the numerator of the simplified -det ToeplitzMatrix((-1/2!,-1),(-1/2!,0))/(det ToeplitzMatrix((-1,1),(-1,-1/2!))*det ToeplitzMatrix((-1,1,0),(-1,-1/2!,0))) = -(1/4)/((3/2)*-2) = 1/12.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A003957.

a(1) = 1; for n > 1, a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=-1, c(2)=-1/2!, c(3)=0, c(4)=1/4!, c(5)=0, c(6)=-1/6!, and c(n) is the coefficient of x^n in the Taylor expansion of cos(x)-x.

a(8)-a(20) from Chai Wah Wu, Feb 10 2024

A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 08 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 2.... A199597
... 1.... 3.... A199598
... 1.... 4.... A199599
... 2.... 1.... A199600
... 2.... 3.... A199601
... 2.... 4.... A199602
... 3.... 0.... A199603, A199604
... 3.... 1.... A199605, A199606
... 3.... 2.... A199607, A199608
... 3.... 3.... A199609, A199610
... 4.... 0.... A199611, A199612
... 4.... 1.... A199613, A199614
... 4.... 2.... A199615, A199616
... 4.... 3.... A199617, A199618
... 4.... 4.... A199619, A199620
... 1.... 0.... A199621
... 1.... 2.... A199622
... 1.... 3.... A199623
... 1.... 4.... A199624
... 2.... 1.... A199625
... 2.... 3.... A199661
... 1.... 0.... A199662
... 1.... 2.... A199663
... 1.... 3.... A199664
... 1.... 4.... A199665
... 2.... 0.... A199666
... 2.... 1.... A199667
... 2.... 3.... A199668
... 2.... 4.... A199669
.. -1.... 0.... A003957
.. -1.... 1.... A199722
.. -1.... 2.... A199721
.. -1.... 3.... A199720
.. -1.... 4.... A199719
.. -2.... 1.... A199726
.. -2.... 2.... A199725
.. -2.... 3.... A199724
.. -2.... 4.... A199723
.. -3.... 1.... A199730
.. -3.... 2.... A199729
.. -3.... 3.... A199728
.. -3.... 4.... A199727
.. -4.... 1.... A199737. A199738
.. -4.... 2.... A199735, A199736
.. -4.... 3.... A199733, A199734
.. -4.... 4.... A199731. A199732
.. -1.... 1.... A199742
.. -1.... 2.... A199741
.. -1.... 3.... A199740
.. -1.... 4.... A199739
.. -2.... 1.... A199776
.. -2.... 3.... A199775
.. -3.... 1.... A199780
.. -3.... 2.... A199779
.. -3.... 3.... A199778
.. -3.... 4.... A199777
.. -4.... 1.... A199782
.. -4.... 3.... A199781
.. -4.... 1.... A199786
.. -4.... 2.... A199785
.. -4.... 3.... A199784
.. -4.... 4.... A199783
.. -3.... 1.... A199789
.. -3.... 2.... A199788
.. -3.... 4.... A199787
.. -2.... 1.... A199793
.. -2.... 2.... A199792
.. -2.... 3.... A199791
.. -2.... 4.... A199790
.. -1.... 1.... A199797
.. -1.... 2.... A199796
.. -1.... 3.... A199795
.. -1.... 4.... A199794
.. -4.... 1.... A199873
.. -4.... 3.... A199872
.. -3.... 1.... A199871
.. -3.... 2.... A199870
.. -3.... 3.... A199869
.. -3.... 4.... A199868
.. -2.... 1.... A199867
.. -2.... 3.... A199866
.. -1.... 1.... A199865
.. -1.... 2.... A199864
.. -1.... 3.... A199863
.. -1.... 4.... A199862
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A199597, take f(x,u,v)=x^2+u*x*cos(x)-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			1.1881851344514388032178109729076525973...

Index entries for transcendental numbers.

Cf. A199370, A199170, A198866, A198755, A198414, A197737, A199429.

(* Program 1:  A199597 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -Pi, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110]
RealDigits[r]  (* A199597 *)
(* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A199597 *)

Edited by Georg Fischer, Aug 03 2021

cp's OEIS Frontend

A212112 Numerators of convergents to the Dottie number, A003957.

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Mathematica

A212113 Denominators of convergents to the Dottie number, A003957.

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Mathematica

A182503 Engel expansion of the Dottie number, A003957.

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Mathematica

A330119 Degrees-based analog to the Dottie number (A003957).

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PARI

A182101 Random walk determined by the binary digits of the Dottie number, A003957.

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Mathematica

A100547 Decimal expansion of -cos(2^19*r) where r ~ 0.739085 is the root of cos x = x (A003957).

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Mathematica

A180619 Consider the function f(n)=1/(Abs(n-r)), where r is the Dottie number, A003957. Let g(n) be defined by the recursion g(n)=Cos(g(n-1)),g(0)=1. Now, a(n)=floor(f(g(n))).

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A369186 The denominators of a series that converges to the Dottie Number (A003957).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A369187 The numerators of a series that converges to the Dottie Number (A003957).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).