A062546 -id:A062546 - cp's OEIS frontend

A138732 Continued fraction for 8th Du Bois Reymond constant.

0, 99232, 2, 6, 1, 3, 4, 2, 1, 2, 1, 16, 8, 2, 57, 13, 2, 1, 16, 1, 1, 6, 5, 5, 1, 1, 6, 1, 1, 3, 1, 1, 2, 9, 18, 1, 8, 15, 2, 2, 1, 2, 1, 2, 1, 5, 2, 3, 5, 1, 3, 4, 17, 11, 1, 2, 1, 6, 1, 2, 3, 15, 3, 12, 1, 8, 6, 1, 1, 1, 2, 4, 29, 44, 1, 1, 7, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 22, 1, 2, 5, 8, 2, 2, 5, 2

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)^4), {x, I}]]], 100] (* Artur Jasinski *)

contfrac((3*exp(8) - 24*exp(6) + 36*exp(4) - 8*exp(2)  - 1167)/384) \\ Michel Marcus, Sep 09 2013

A138733 Second term of continued fraction for 2n-th Du Bois Reymond constant.

Original entry on oeis.org

5, 190, 4531, 99232, 2125044, 45190209, 958768567, 20325471335, 430773893366, 9128872855695, 193450867955197, 4099389985205820, 86869246502331992, 1840823999333339814, 39008411877876819180, 826616742911186406242

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

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

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

a(n) = floor(1/C(2n)), where C(2n) is the 2n-th Du Bois Reymond constant. [From Max Alekseyev, Sep 15 2009]

Extended by Max Alekseyev, Sep 15 2009

A255272 Decimal expansion of the second smallest positive root of tan(x) = x.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 20 2015

This constant is quite close to 5*Pi/2 - 1/8 = 7.72898...

Searching for solutions x=k*Pi+Pi/2-e and small e, for k=1,2,3.... means via the approximation tan(x) = 1/e-e/3-e^3/45... that e is approximately 1/(k*Pi+Pi/2), so the constants x are close to k*Pi+Pi/2-1/(k*Pi+Pi/2). Here k=2 and the constant is close to 5*Pi/2-2/(5*Pi) = 7.7266576... - R. J. Mathar, Jul 11 2024

			7.72525183693770716419506893306298662637815930461...

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

Cf. A115365 (smallest positive root), A062546 (C_2 = 2nd du Bois-Reymond constant), A224196 (C_3), A207528 (C_4), A243108 (C_5), A245333 (C_6).

Cf. A079330, A088989.

xi[n_] := x /. FindRoot[Tan[x] == x, {x, n*Pi + Pi/2 - 1/(4*n)}, WorkingPrecision -> 102]; RealDigits[xi[2]] // First

solve(x=7,7.8,tan(x)-x) \\ Charles R Greathouse IV, Apr 20 2016

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).

Original entry on oeis.org

0, 1, 0, 1, -1, -1, -1, 0, 0, 3, 1, -5, 18, -13, -7, -11, 70, -135, 65, -10, 45, 111, -609, 1215, -1350, 1275, -621, -141, -1009, 6188, -16758, 27335, -26845, 12474, -2548, 1883, 10977, -81353, 270004, -511791, 584710, -420287, 216468, -70169, -3599, -146691, 1248210, -4715217, 10303461, -14439411

Offset: 0

Gerald McGarvey, Mar 02 2005

For n>0 the row sums = (-1)^(n-1) * (n-1)! For n odd, the sum of the absolute values of the coefficients in the n-th row = (2*(n-1))!/n! (every other entry of A001761).
The sum of the (2n)th du Bois-Reymond constants = 1/5 or is very close to 1/5.
For the 6th and 9th rows, the coefficients were adjusted from results of the residue evaluations so that double factorials ((2n)!! = 2^n*n! (A000165)) are in the denominators. For the 6th row they were multiplied by 3, for the 9th row they were multiplied by 9.
For n>1, Sum_{k=0..n} (n-k+1)*a(n,k) = (-1)^(n)*A001286(n-1) [A001286 are Lah numbers: (n-1)*n!/2].

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

Cf. A000142, A000165, A001761, A062545, A062546, A085466, A085467.

Table[2 Residue[x^2/((1+x^2)^n (Tan[x]-x)), {x, I}], {n, 0, 9}]

For n>1, C_2n = -3 - 2 * Residue_{x=i} (x^2/((1+x^2)^n * (tan(x) - x))) (see MathWorld article).

For n>1, Sum_{k=0..n} (-1)^(n+k)*a(n, k) = (2*(n-1))!/n! (i.e., A001761(n-1)).

Added the keyword tabl Gerald McGarvey, Aug 20 2009

A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).

Original entry on oeis.org

1, 4, 0, 8, 5, 9

Offset: 0

Bernard Schott, Apr 23 2021

The equation of the sinc function is y = sin(x)/x.
Equivalently, sum of f(x) = sinc(x) where x > 0 and f'(x) = 0. - David A. Corneth, May 01 2021
These extreme values are obtained when x_k > 0 is a solution to tan(x) = x (see Chronomath link), or equivalently to y = tanc(x) = tan(x)/x = 1. The corresponding k-th extreme value is y_k = sin(x_k)/x_k.
Every extremum y_k = (-1)^k/(k*Pi) + O(1/k^2), hence the series Sum_{k > 0} sin(x_k)/x_k is convergent.
However, this series is not absolutely convergent, just as (C_1)/2 diverges where C_1 is the corresponding du Bois-Reymond constant.

			-0.140859...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.18, pp. 285 and 303.

Serge Mehl, Comportement en zéro de sin(x)/x, ChronoMath.
Eric Weisstein's World of Mathematics, du Bois-Reymond constants.
Eric Weisstein's World of Mathematics, Sinc Function.
Eric Weisstein's World of Mathematics, Tanc Function.

Coordinates of the 1st extremum: A115365 (x_1), A213053 (y_1).

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

Equals Sum_{k >= 1} sinc(x_k) or Sum_{k >= 1} (-1)^k / sqrt(1+(x_k)^2), where x_k is the k-th positive root of x = tan(x).

More terms from Amiram Eldar, Apr 23 2021

Name clarified by N. J. A. Sloane, May 01 2021

cp's OEIS Frontend

A138732 Continued fraction for 8th Du Bois Reymond constant.

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A138733 Second term of continued fraction for 2n-th Du Bois Reymond constant.

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A255272 Decimal expansion of the second smallest positive root of tan(x) = x.

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A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)*tanh(1)^k) / (2^n*n! * (1-tanh(1))^n).

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Programs

Mathematica

Formula

Extensions

A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).

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Author

Keywords

Comments

Examples

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Links

Crossrefs

Formula

Extensions

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).