cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A009379 Expansion of log(1+tan(x)*x).

Original entry on oeis.org

0, 2, -4, 96, -2080, 125440, -8629248, 996007936, -140162633728, 27058965184512, -6350990843576320, 1866805063173799936, -653569786506324738048, 273136898848234632380416, -133034893921204302732328960
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A009399, A085984.

Programs

  • Mathematica
    Log[ 1+Tan[ x ]*x ] (* Even Part *)
nn = 20; Table[(CoefficientList[Series[Log[1 + x*Tan[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Jan 23 2015 *)

Formula

a(n) ~ (2*n)! * (-1)^(n+1) / (n * r^(2*n)), where r = 1.1996786402577338339163698486411419442614587884... (see A085984) is the root of the equation r*tanh(r) = 1. - Vaclav Kotesovec, Jan 23 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

A069855 Decimal expansion of the root of x*tan(x)=1.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, May 01 2002

Keywords

Comments

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and the points A = (u, v), B = (-u, v), C = (-u, -v), D = (u, -v), K = (0, 2v), L = (-2u, 0), M = (0, -2v), N = (2u,0), where u is given by this sequence, and v = u/sqrt(1+u^2). Then ABCD is the rectangle of maximal area, inscribed in S, with sides parallel to the coordinate axes, and KLMN is the rhombus of minimal area, circumscribed around S, with vertices on the coordinate axes. Also, A,B,C,D are the tangent points where the sides of the rhombus touch S, see illustration in the links section. - Gleb Koloskov, Jul 05 2021

Examples

			0.860333589019379762483893424137662333411884363237653783...

Links

Crossrefs

Cf. A009399, A085984, A346062.

Programs

  • Mathematica
    N[Minimize[{(x+Cot[x])^2 Sin[x],{x>0,xGleb Koloskov, Jul 05 2021 *)
RealDigits[x/.FindRoot[x Tan[x]==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 04 2021 *)
  • PARI
    /* 300 significant digits */ s=0.1; for(n=1,500,s=s+sign(cotan(s)-s)/2^n; if(n>499, print(s*1.)))

Formula

Equals A346062 * sqrt(2 + 2*sqrt(1 + 256/A346062^2)) / 16. - Gleb Koloskov, Jul 05 2021

A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -2, 18, -312, 9470, -436860, 28616322, -2522596496, 288046961190, -41355026494020, 7291524732108650, -1548849359704927896, 390122366308850972238, -114968364853645904762252, 39189956630839558368115410, -15300235972710835734174638880
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			log(1 + x*tan(x/2)) = x^2/2! - 2*x^4/4! + 18*x^6/6! - 312*x^8/8! + ...

Links

Crossrefs

Cf. A001469, A003707, A009379, A009399, A110501, A296838.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tan(x/2)).
a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.54340463841820844795870974005331555369788376471926269... is the root of the equation r*tanh(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -4, 48, -1186, 50060, -3226206, 294835184, -36270477034, 5779302944436, -1157856177719830, 284876691727454552, -84442374415240892898, 29680054107768128647388, -12205478262363331593956686, 5805823539844285054558025280, -3163004294186696659107788567386
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

Links

Crossrefs

Cf. A001469, A003707, A009379, A009399, A110501, A296837.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tanh(x/2)).
a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A024295 Expansion of log(1+tanh(x)*x)/2.

Original entry on oeis.org

0, 1, -10, 288, -16656, 1629440, -242569728, 51150403584, -14514466195456, 5334038568566784, -2464645606175539200, 1398530065161670098944, -956068708752272063987712, 775009022325905883458961408, -735038625130292891682693185536
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A009399.

Programs

  • Mathematica
    With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]*x]/2,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Mar 04 2013 *)

Extensions

Extended with signs Mar 1997
Showing 1-5 of 5 results.

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