A062546 -id:A062546 - cp's OEIS frontend

A115365 Decimal expansion of smallest positive root of tan(x) = x.

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

Location (for x>0) of the first negative lobe of sinc(x) = sin(x)/x, where sinc(x) attains its absolute minimum of -0.217233628... The function sinc(x) is important in spectral theory (transient data truncation artifacts). - Stanislav Sykora, Mar 05 2012
Also the first root of the sinc(3,x) function, that is, the radial component of the 3D Fourier transform of 3-dimensional unit sphere. Also the first root of the spherical Bessel function of the 1st kind, j_1(x). - Stanislav Sykora, Nov 14 2013
Unique fixed point of the function arctan(x)+Pi, and this fixed point is attractive. - Robert FERREOL, May 09 2023
Further roots (intersections of y=x with other branches of y=tan(x)) are at x=7.725251... = A255272, x=10.9041216..., x=14.0661939..., x= 17.2207552.. etc. - R. J. Mathar, Jul 11 2024

			4.4934094579090641753...

M. Abramowitz, I. A. Stegun, Editors, Handbook of Mathematical Functions, Dover Publications, 1965, Chapter 10.

G. C. Greubel, Table of n, a(n) for n = 1..5000
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.
Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60.
Mohammad K. Azarian, Solution to Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal Vol. 39, No. 1, January 2008, pp. 66-67.
Stanislav Sykora, K-Space Images of n-Dimensional Spheres and Generalized Sinc Functions
Eric Weisstein's World of Mathematics, Tangent
Eric Weisstein's World of Mathematics, Tanc Function
Index entries for transcendental numbers

Cf. A102015 (continued fraction), A213053 (amplitude at x).

Cf. A062546, A224196, A207528, A243108, A245333.

Digits:=200; fsolve(x*cos(x)-sin(x),x,4..5);

RealDigits[FindRoot[Tan[x]==x,{x,4}, WorkingPrecision->128][[1,2]]][[1]] (* Robert G. Wilson v, Mar 05 2012; corrected by Harvey P. Dale, Mar 22 2012 *)
RealDigits[BesselJZero[3/2, 1], 10, 100][[1]] (* Vladimir Reshetnikov, May 13 2016 *)

solve(x=4,4.5,tan(x)-x) \\ Charles R Greathouse IV, Jun 10 2012

besseljzero(3/2,1) \\ Charles R Greathouse IV, Jan 23 2025

A062545 Continued fraction for the 2nd du Bois-Reymond constant.

Original entry on oeis.org

0, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 0

Jason Earls, Jun 26 2001

Francois Le Lionnais, Les nombres remarquables, Paris, Hermann 1983, pp. 23.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A062546.

Join[{0}, LinearRecurrence[{2, -1}, {5, 7}, 100]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
ContinuedFraction[(E^2 - 7)/2, 40] (* Alonso del Arte, Feb 20 2012 *)

PARI
```
contfrac((exp(2)-7)/2)
```

A224196 Decimal expansion of the 3rd du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 15 2013

From Jon E. Schoenfield, Aug 17 2014: (Start)
Evaluating the partial sums
   Sum_{k=1..j} 2*(1 + x_k ^ 2)^(-3/2)
(where x_k is the k-th root of tan(t)=t; see the Mathworld link) at j = 1, 2, 4, 8, 16, 32, ..., it becomes apparent that they approach
   c0 + c2/j^2 + c3/j^3 + c4/j^4 + ...
where
   c0 = 0.02825176416006793787321073299629898515427...
and c2 and c3 are -4/Pi^3 and 16/Pi^3, respectively.
(The k-th root of tan(t)=t is
   r - d_1/r - d_2/r^3 - d_3/r^5 - d_4/r^7 - d_5/r^9 - ...
where r = (k+1/2) * Pi and d_j = A079330(j)/A088989(j).) (End)
d_n = A079330(n)/A088989(n) ~ Gamma(1/3) / (2^(2/3) * 3^(1/6) * Pi^(5/3)) * (Pi/2)^(2*n) / n^(4/3). - Vaclav Kotesovec, Aug 19 2014

			0.028251764...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.

Jon E. Schoenfield and Vaclav Kotesovec, Table of n, a(n) for n = 0..450 (first 100 terms from Jon E. Schoenfield)
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A062546 (2nd), A207528 (4th), A243108 (5th), A245333 (6th).

digits = 16; m0 = 10^5; dm = 10^5; Clear[xi, c3]; xi[n_?NumericQ] := xi[n] = x /. FindRoot[x == Tan[x], {x, n*Pi + Pi/2 - 1/(4*n)}, WorkingPrecision -> digits + 5]; c3[m_] := c3[m] = 2*Sum[1/(1 + xi[n]^2)^(3/2), {n, 1, m}] - 2*PolyGamma[2, m + 1]/(2*Pi^3); c3[m0] ; c3[m = m0 + dm]; While[RealDigits[c3[m], 10, digits] != RealDigits[c3[m - dm], 10, digits], Print["m = ", m, " ", c3[m]]; m = m + dm]; RealDigits[c3[m], 10, digits] // First

a(8)-a(15) from Robert G. Wilson v, Nov 06 2013

More terms from Jon E. Schoenfield, Aug 17 2014

A085466 a(n) is the denominator of the polynomial in e^2 giving the (2n)th du Bois Reymond constant.

Original entry on oeis.org

2, 8, 32, 384, 1536, 10240, 368640, 10321920, 4587520, 297271296, 29727129600, 435997900800, 15695924428800, 116598295756800, 1523551064555520, 1371195958099968000, 5484783832399872000, 41440588955910144000

Offset: 1

Eric W. Weisstein, Jul 01 2003

nonn

			{(-7 + e^2)/2, (-25 - 4*e^2 + e^4)/8, (-98 + 3*e^2 - 6*e^4 + e^6)/32}

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A085467.

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a := proc(n) local r ; r := residue(x^2/(1+x^2)^n/(tan(x)-x),x=I) ; r := -3-2*subs(tanh(1)=(x-1/x)/(x+1/x),%) ; r := taylor(r,x=0,16*n+2) ; cf := 1 ; for p from 0 to 2*n by 2 do cf := lcm(cf,denom(coeftayl(r,x=0,p))) ; od ; r := simplify(convert(r*cf,polynom)) ; RETURN([cf,r]) ; end: A085466 := proc() # n = 1 invalid formula printf("2, ") ; for n from 2 to 14 do a085467 := a(n)[1] : printf("%d, ",a085467) ; od : end: A085466() ; # R. J. Mathar, Apr 05 2007

a = {}; Do[p = FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x, I}]]]; AppendTo[a, Denominator[p]], {n, 1, 9}]; a (* Artur Jasinski, Mar 26 2008 *)

More terms from R. J. Mathar, Apr 05 2007

Extended by Max Alekseyev, Sep 15 2009

A207528 Decimal expansion of 4th du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Feb 24 2012

			0.0052407046777047711...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A138730 (continued fraction expansion).

Cf. A062546 (c2), A224196 (c3), A243108 (c5), A245333 (c6).

RealDigits[(E^4 - 4E^2 - 25)/8, 10, 100][[1]]

2*Sum_{k>=1} (1 + (x_k)^2)^(-2) = (e^4 - 4*e^2 - 25)/8, where x_k is the k-th root of t = tan t and e = 2.71828... is the natural logarithm base.

A243108 Decimal expansion of the 5th du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 19 2014

The k-th Du Bois Reymond constant c_k is asymptotic to 2/(1+r^2)^(k/2), where r = A115365 = 4.4934094579090641753... is smallest positive root of the equation tan(r) = r. - Vaclav Kotesovec, Aug 20 2014

			0.00105610210733480920562199158210781176744608061...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.

Vaclav Kotesovec, Table of n, a(n) for n = 0..450
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A062546 (c2), A224196 (c3), A207528 (c4), A245333 (c6).

Cf. A115365, A079330, A088989.

A138730 Continued fraction for 4th Du Bois Reymond constant.

Original entry on oeis.org

0, 190, 1, 4, 2, 1, 1, 1, 6, 10, 6, 6, 1, 3, 2, 9, 67, 2, 3, 1, 7, 1, 2, 1, 1, 1, 2, 1, 1, 4, 1, 68, 5, 6, 1, 1, 1, 4, 1, 1, 5, 1, 4, 8, 2, 5, 5, 1, 1, 3, 1, 2, 2, 6, 1, 1, 1, 9, 2, 1, 1, 1, 17, 5, 21, 2, 9, 1, 3, 1, 2, 4, 1, 3, 5, 1, 56, 1, 4, 14, 5, 17, 4, 2, 34, 1, 18, 1, 1, 4, 1, 5, 16, 1, 3, 27, 1, 11

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A207528, A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^2), {x, I}]]], 100](*Artur Jasinski*)

contfrac((exp(4)-4*exp(2)-25)/8) \\ Charles R Greathouse IV, Feb 24 2012

A245333 Decimal expansion of the 6th du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 18 2014

			0.000220674708176213169277990873376019646763182409385382227156501293654...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.12 Du Bois Reymond Constants, p. 238.

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants

Cf. A062546 (c2), A224196 (c3), A207528 (c4), A243108 (c5).

c6 = (1/32)*(E^6 - 6*E^4 + 3*E^2 - 98); Join[{0, 0, 0}, RealDigits[c6, 10, 102] // First]

c_6 = (1/32)*(e^6 - 6*e^4 + 3*e^2 - 98).

A138729 a(n) = -A085466(n) times the free coefficient of the 2n-th Du Bois-Reymond polynomial in e^2.

Original entry on oeis.org

7, 25, 98, 1167, 4650, 30930, 1111860, 31100895, 13812610, 894570642, 89419472100, 1311049104750, 47185076099700, 350440263072900, 4578242813103960, 4119778157653533375, 16476927824724617250, 124478128839848033250, 40326914169455544130500

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

a = {7}; Do[p = FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^n), {x, I}]]]; AppendTo[a, -First[p[[2]]]], {n, 2, 9}]; a (*Artur Jasinski*)

Edited and extended by Max Alekseyev, Sep 15 2009

A138731 Continued fraction for 6th Du Bois Reymond constant.

Original entry on oeis.org

0, 4531, 1, 1, 3, 1, 8, 3, 2, 1, 3, 1, 1, 9, 1, 2, 9, 1, 1, 5, 3, 2, 1, 1, 5, 1, 1, 3, 5, 3, 2, 1, 6, 1, 7, 2, 4, 1, 3, 1, 23, 2, 8, 2, 1, 19, 1, 23, 1, 3, 1, 18, 1, 2, 1, 1, 3, 24, 1, 4, 1, 1, 5, 1, 1, 2, 1, 1, 89, 1, 1, 1, 2, 1, 8, 1, 52, 1, 1, 1, 1, 1, 4, 1, 15, 1, 1, 1, 5, 3, 1, 2, 1, 4, 1, 4, 24, 5, 1, 5

Offset: 1

Artur Jasinski, Mar 26 2008

Cf. A062545, A062546, A085466, A085467, A138729, A138730, A138731, A138732, A138733.

ContinuedFraction[FullSimplify[TrigToExp[ -3 - 2Residue[x^2/((Tan[x] - x) (1 + x^2)^3), {x, I}]]], 100](*Artur Jasinski*)

contfrac((exp(6)-6*exp(4)+3*exp(2)-98)/32) \\ Charles R Greathouse IV, Feb 24 2012

cp's OEIS Frontend

A115365 Decimal expansion of smallest positive root of tan(x) = x.

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A062545 Continued fraction for the 2nd du Bois-Reymond constant.

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A224196 Decimal expansion of the 3rd du Bois-Reymond constant.

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A085466 a(n) is the denominator of the polynomial in e^2 giving the (2n)th du Bois Reymond constant.

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A207528 Decimal expansion of 4th du Bois-Reymond constant.

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A243108 Decimal expansion of the 5th du Bois-Reymond constant.

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A138730 Continued fraction for 4th Du Bois Reymond constant.

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A245333 Decimal expansion of the 6th du Bois-Reymond constant.

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