A088989 -id:A088989 - cp's OEIS frontend

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

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

A088306 Integers n with tan n > |n|, ordered by |n|.

Original entry on oeis.org

1, -2, -11, -33, -52174, 260515, -573204, 37362253, -42781604, 122925461, 534483448, 3083975227, 902209779836, -2685575996367, -65398140378926, 74357078147863, 214112296674652, 642336890023956, -5920787228742393, -12055686754159438, 18190586279576483, -48436859313312404

Offset: 1

Paul Boddington, Nov 05 2003

sign

Name was "Positive integers n with |tan n| > n." before signs were added. The sign here shows whether tan(|n|) is positive or negative.
That this sequence is infinite was proved by Bellamy, Lagarias and Lazebnik. It seems not to be known whether there are infinitely many n with tan n > n.
At approximately 2.37e154, there is a value of n which has tan(n)/n > 556. - Phil Carmody, Mar 04 2007 [This is index 214 in the b-file.]
As n increases, log(|a(n)|)/n seems to approach Pi/2; this is similar to what would be expected if an integer sequence were created by drawing many random numbers independently from a uniform distribution on the interval [-Pi/2,+Pi/2] and including in the sequence only those integers j for which the j-th random number x_j happened to satisfy |x_j| < 1/j (and applying to j the sign of x_j). - Jon E. Schoenfield, Aug 19 2014; updated Nov 07 2014 to reflect the change in the sequence's Name)

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
D. Bellamy, J. C. Lagarias and F. Lazebnik, Proposed problem: large values of Tan n
Jon E. Schoenfield, Magma program

Cf. A000503, A224269, A079330, A088989.

Cf. A249836 (subsequence of positive terms).

a:=proc(n) if abs(evalf(tan(n)))>n then n else fi end: seq(a(n),n=1..100000); # Emeric Deutsch, Dec 18 2004

Select[Range[600000],Abs[Tan[#]]>#&] (* Harvey P. Dale, Nov 30 2012 *)

is(n)=tan(n)>abs(n) \\ Charles R Greathouse IV, Nov 07 2014

More terms from Jon E. Schoenfield, Aug 17 2014

Signs added and other edits by Franklin T. Adams-Watters, Sep 09 2014

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.

A079330 Numerators of coefficients of odd powers of 1/q in the solution series for tan(x)/x=1.

Original entry on oeis.org

1, 2, 13, 146, 781, 16328, 6316012, 38759594, 9655714457, 50134571594, 25626917879, 638499558282328, 125381104727404588, 435948294065152496, 146414084312394268792, 1076603090723736731974978

Offset: 1

Eric W. Weisstein, Jan 03 2003

Series contributed by David W. Cantrell.

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, equation 34:7:4 at page 325.

Vaclav Kotesovec, Table of n, a(n) for n = 1..250
Vaclav Kotesovec, Asymptotic of the coefficients A079330 / A088989.
Eric Weisstein's World of Mathematics, Tanc Function.

Denominators are in A088989.

Cf. A224196.

Last/@Partition[CoefficientList[InverseSeries[Series[x+Cot[x], {x, 0, 50}], q], 1/q], 2]

a(n)/A088989(n) ~ c * (Pi/2)^(2*n) / n^(4/3), where c = GAMMA(1/3)/(2^(2/3)*3^(1/6)*Pi^(5/3)) = 0.208532... . - Vaclav Kotesovec, Aug 19 2014

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A088306 Integers n with tan n > |n|, ordered by |n|.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A079330 Numerators of coefficients of odd powers of 1/q in the solution series for tan(x)/x=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI