A003957 -id:A003957 - cp's OEIS frontend

A332506 Decimal expansion of the number snc(2 Pi), where snc is the sine-normal-to-cosine function; see A332500.

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

Offset: 1

Clark Kimberling, May 05 2020

			snc(2 Pi) = 5.544100173964425835269974678885...

Cf. A332500, A003957, A332523 (numerators of convergents), A332524 (denominators of convergents).

u = u /. FindRoot[ u + Cos[u] == 2 Pi, {u, 0}, WorkingPrecision -> 150]
RealDigits[u][[1]]

snc(2 Pi) = 2 Pi + snc(0), where snc(0) = Dottie number (A003957).

A335563 Decimal expansion of the real part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 14 2020

			0.96226600633306068948508092593102537827547141928666...

T. H. Miller, On the numerical values of the roots of the equation cos x = x, Proc. Edinburgh Math. Soc., Vol. 9 (1890), pp. 80-83.
T. Hugh Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc., Vol. 21 (1902), pp. 160-162 (the last 3 pages of the pdf file).
Eric Weisstein's World of Mathematics, Dottie Number.
Wikipedia, Dottie number.

Cf. A003957, A335564 (the imaginary part), A335565, A335566.

z = {x, y} /. FindRoot[{x == Cos[x]*Cosh[y], y == Sin[x]*Sinh[y]}, {{x, 1}, {y, 1}}, WorkingPrecision -> 100]; RealDigits[z[[1]], 10, 90][[1]]

A335564 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 14 2020

			1.11097438808469174490088735849365794466180028421014...

T. H. Miller, On the numerical values of the roots of the equation cos x = x, Proc. Edinburgh Math. Soc., Vol. 9 (1890), pp. 80-83.
T. Hugh Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc., Vol. 21 (1902), pp. 160-162 (the last 3 pages of the pdf file).
Eric Weisstein's World of Mathematics, Dottie Number.
Wikipedia, Dottie number.

Cf. A003957, A335563 (the real part), A335565, A335566.

z = {x, y} /. FindRoot[{x == Cos[x]*Cosh[y], y == Sin[x]*Sinh[y]}, {{x, 1}, {y, 1}}, WorkingPrecision -> 100]; RealDigits[z[[2]], 10, 90][[1]]

A335565 Decimal expansion of the real part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 14 2020

			5.86956037734761439408817256819284864079702760098009...

Henry E. Fettis, Complex Roots of sin z = az, cos z = az, and cosh z = az, Mathematics of Computation, Vol. 30, No. 135 (1976), pp. 541-545, Table 18, alternative link (without the tables).
T. H. Miller, On the numerical values of the roots of the equation cos x = x, Proc. Edinburgh Math. Soc., Vol. 9 (1890), pp. 80-83.
T. Hugh Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc., Vol. 21 (1902), pp. 160-162 (the last 3 pages of the pdf file).
Eric Weisstein's World of Mathematics, Dottie Number.
Wikipedia, Dottie number.

Cf. A003957, A335563, A335564, A335566 (the imaginary part).

z = {x, y} /. FindRoot[{x == Cos[x]*Cosh[y], y == -Sin[x]*Sinh[y]}, {{x, 5}, {y, 2}}, WorkingPrecision -> 100]; RealDigits[z[[1]], 10, 90][[1]]

A335566 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 14 2020

			2.54488576688570932655597042567309972354822580168081...

Henry E. Fettis, Complex Roots of sin z = az, cos z = az, and cosh z = az, Mathematics of Computation, Vol. 30, No. 135 (1976), pp. 541-545, Table 18, alternative link (without the tables).
T. H. Miller, On the numerical values of the roots of the equation cos x = x, Proc. Edinburgh Math. Soc., Vol. 9 (1890), pp. 80-83.
T. Hugh Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc., Vol. 21 (1902), pp. 160-162 (the last 3 pages of the pdf file).
Eric Weisstein's World of Mathematics, Dottie Number.
Wikipedia, Dottie number.

Cf. A003957, A335563, A335564, A335565 (the real part).

z = {x, y} /. FindRoot[{x == Cos[x]*Cosh[y], y == -Sin[x]*Sinh[y]}, {{x, 5}, {y, 2}}, WorkingPrecision -> 100]; RealDigits[z[[2]], 10, 90][[1]]

A336083 Decimal expansion of the arclength on the unit circle such that the corresponding chord separates the interior into segments having 3 = ratio of segment areas; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 11 2020

Suppose that s in (0,Pi) is the length of an arc of the unit circle. The associated chord separates the interior into two segments. Let A1 be the area of the larger and A2 the area of the smaller. The term "ratio of segment areas" means A1/A2. See A336073 for a guide to related sequences.

Equals the median of the probability distribution function of angles of random rotations in 3D space uniformly distributed with respect to the Haar measure, i.e., the solution x to Integral_{t=0..x} ((1 - cos(t))/Pi) dt = 1/2 (see Reynolds, 2017; cf. A086118, A361605). - Amiram Eldar, Mar 17 2023

			arclength = 2.3098814600100572608866337793136248461119964...

Marc B. Reynolds, Volume element of SO(3) and average uniform random rotation angle, 2017.

Cf. A336073, A003957, A019669.

Cf. A086118, A361605.

k = 3; s = s /. FindRoot[(2 Pi - s + Sin[s])/(s - Sin[s]) == k, {s, 2}, WorkingPrecision -> 200]
RealDigits[s][[1]]

d=solve(x=0,1,cos(x)-x); d+Pi/2 \\ Gleb Koloskov, Feb 21 2021

Equals d+Pi/2 = A003957 + A019669, where d is the Dottie number. - Gleb Koloskov, Feb 21 2021

A009442 E.g.f. log(1 + x/cos(x)).

Original entry on oeis.org

0, 1, -1, 5, -18, 109, -720, 5977, -56336, 612729, -7453440, 100954061, -1502172672, 24395453861, -429076910080, 8128143367905, -164961704478720, 3571195811862385, -82142328351817728, 2000535014776893973

Offset: 0

R. H. Hardin

Cf. A003957.

CoefficientList[Series[Log[1 + x*Sec[x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)

a(n):=2*n!*sum(((-1)^(n-1)*sum(binomial((n/2-m+j-1),j)*4^(m-j)*sum((i-j)^(2*m)*binomial(2*j,i)*(-1)^(m+j-i),i,0,j),j,0,m))/((n-2*m)*(2*m)!),m,1,(n-1)/2)+(-1)^(n-1)*n!/n; /* Vladimir Kruchinin, Jun 16 2011 */

a(n)=2*n!*sum(m=1..(n-1)/2, ((-1)^(n-1)*sum(j=0..m, binomial((n/2-m+j-1),j)*4^(m-j)*sum(i=0..j, (i-j)^(2*m)*binomial(2*j,i)*(-1)^(m+j-i))))/((n-2*m)*(2*m)!))+(-1)^(n-1)*n!/n. - Vladimir Kruchinin, Jun 16 2011

a(n) ~ (n-1)! * (-1)^(n+1) / r^n, where r = 0.7390851332151606416553120876738734040134117589... (see A003957) is the root of the equation cos(r) = r. - Vaclav Kotesovec, Jan 24 2015

Extended with signs by Olivier Gérard, Mar 15 1997

A197003 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=0.3695425666075803208276560438369...
yo=0.4039727532995172093189617400663...
m=1.09316974498501692208815321416057...
|OP|=0.54749949218543621432520415035...

Cf. A003957, A197002, A196996, A197000.

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]

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

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

A198822 Decimal expansion of x > 0 satisfying x^2 - 2*cos(x) = 2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 30 2011

See A198755 for a guide to related sequences. The Mathematica program includes a graph.

			1.47817026643032128331062417534774680802682351780...

Index entries for transcendental numbers.

Cf. A003957, A198755.

a = 1; b = -2; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r] (* A198822 *)

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

Equals 2*A003957. - Gleb Koloskov, Jun 16 2021

A217066 E.g.f. is series reversion of x*(sec(x)+tan(x)).

Original entry on oeis.org

1, -2, 9, -68, 725, -9966, 167629, -3334120, 76543785, -1992009850, 57948521521, -1863394764780, 65631109286717, -2512768138160294, 103905545328667125, -4615035074291158352, 219122841820491458897, -11075488610594107402098, 593746153204862664363481

Offset: 1

Vladimir Kruchinin, Sep 26 2012

sign

Cf. A003957.

a[n_] := Sum[ (-1)^k*Binomial[n+k-1, n-1] * Sum[ Binomial[n-2*i-2, k-1] * Sum[ (-1)^(j+i)*2^(-n+k-j+2*i+1)*StirlingS2[n-1, n+j-2*i-1] * Binomial[n+j-2*i-2, n-2*i-2]*(n+j-2*i-1)!, {j, 0, 2*i}], {i, 0, (n-k-1)/2}], {k, 1, n-1}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Feb 22 2013 *)
Rest[CoefficientList[InverseSeries[Series[x*(Sec[x]+Tan[x]), {x, 0, 20}], x],x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 22 2014 *)

a(n):=if n=1 then 1 else
sum((-1)^k*binomial(n+k-1,n-1)*sum(binomial(n-2*i-2,k-1)*sum((-1)^(j+i)*2^(-n+k-j+2*i+1)*stirling2(n-1,n+j+(-2)*i-1)*binomial(n+j+(-2)*i-2,n-2*i-2)*(n+j+(-2)*i-1)!,j,0,2*i),i,0,(n-k-1)/2),k,1,n-1);

a(n) = sum(k=1..n-1, (-1)^k*binomial(n+k-1,n-1)*sum(k=1..n-1, binomial(n-2*i-2,k-1)*sum(i=0..(n-k-1)/2, (-1)^(j+i)*2^(-n+k-j+2*i+1)*stirling2(n-1,n+j+(-2)*i-1)*binomial(n+j+(-2)*i-2,n-2*i-2)*(n+j+(-2)*i-1)!,j,0,2*i))), n>1, a(1)=1.

a(n) ~ (-1)^(n+1) * n^(n-1) * s / (sqrt(1+sin(s)) * exp(n) * (1-sin(s))^n), where s = 0.73908513321516... (see A003957) is the root of the equation s = cos(s). - Vaclav Kotesovec, Jan 22 2014

cp's OEIS Frontend

A332506 Decimal expansion of the number snc(2 Pi), where snc is the sine-normal-to-cosine function; see A332500.

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A335563 Decimal expansion of the real part of the complex root of cos(x + i*y) = x - i*y with least x > 0 and y > 0.

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A335564 Decimal expansion of the imaginary part of the complex root of cos(x + i*y) = x - i*y with least x > 0 and y > 0.

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A335565 Decimal expansion of the real part of the complex root of cos(x + i*y) = x + i*y with least x > 0 and y > 0.

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A335566 Decimal expansion of the imaginary part of the complex root of cos(x + i*y) = x + i*y with least x > 0 and y > 0.

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A336083 Decimal expansion of the arclength on the unit circle such that the corresponding chord separates the interior into segments having 3 = ratio of segment areas; see Comments.

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PARI

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A009442 E.g.f. log(1 + x/cos(x)).

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Mathematica

Maxima

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Extensions

A197003 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).

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Comments

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Programs

Mathematica

PARI

Formula

A198822 Decimal expansion of x > 0 satisfying x^2 - 2*cos(x) = 2.

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A335563 Decimal expansion of the real part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

A335564 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

A335565 Decimal expansion of the real part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.

A335566 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.