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 1-6 of 6 results.

A295254 Expansion of e.g.f. csch(x)*(1 - sqrt(1 - 4*sinh(x)))/2.

Original entry on oeis.org

1, 1, 4, 31, 352, 5341, 101824, 2341291, 63092992, 1950837241, 68093599744, 2648776394551, 113633946898432, 5330308817264341, 271416230974603264, 14910196369733535811, 879003840976919068672, 55354496206857969062641, 3708594029795800700944384, 263391744037123969891925071
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A136630, A295237, A295255, A295256, A295257, A295258.

Programs

  • Maple
    a:=series(csch(x)*(1-sqrt(1-4*sinh(x)))/2,x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Csch[x] (1 - Sqrt[1 - 4 Sinh[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sinh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - sinh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2) * 17^(1/4) * n^(n-1) / (exp(n) * (log((1+ sqrt(17))/4))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} k! * binomial(2*k+1,k)/(2*k+1) * A136630(n,k). - Seiichi Manyama, Feb 23 2025

A295256 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*cosh(x))).

Original entry on oeis.org

1, 1, 4, 33, 384, 5945, 115680, 2713417, 74568704, 2350925649, 83660474880, 3317599815761, 145087264278528, 6937450761100873, 360078818344534016, 20162761727269502265, 1211588127198611374080, 77769423447774393465377, 5310706204624302598127616, 384439720034220718046773249
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A185951, A295237, A295254, A295255, A295257, A295258.

Programs

  • Maple
    a:=series(2/(1+sqrt(1-4*x*cosh(x))),x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Cosh[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Cosh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2 + 2*r*sqrt(1-16*r^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.2428073624074744554637516823... is the root of the equation 2*r*(exp(2*r)+1) = exp(r). - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} k! * binomial(2*k+1,k)/(2*k+1) * A185951(n,k). - Seiichi Manyama, Feb 23 2025

A196605 Decimal expansion of the least x>0 satisfying sec(x)=4x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 04 2011

Keywords

Examples

			0.25859858225418949030448261956152028133855...

Links

Crossrefs

Cf. A295255.

Programs

  • Mathematica
    Plot[{1/x, Cos[x], 2 Cos[x], 3*Cos[x], 4 Cos[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, .1, 5}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == 2 Cos[x], {x, .5, .7}, WorkingPrecision -> 100]
RealDigits[t]  (* A196603 *)
t = x /. FindRoot[1/x == 3 Cos[x], {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196604 *)
t = x /. FindRoot[1/x == 4 Cos[x], {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196605 *)
t = x /. FindRoot[1/x == 5 Cos[x], {x, .15, .23}, WorkingPrecision -> 100]
RealDigits[t]  (* A196606 *)
t = x /. FindRoot[1/x == 6 Cos[x], {x, .1, .2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196607 *)

A295257 Expansion of e.g.f. cot(x)*(1 - sqrt(1 - 4*tan(x)))/2.

Original entry on oeis.org

1, 1, 4, 32, 368, 5656, 109024, 2533712, 68995328, 2155513216, 76014982144, 2987332904192, 129473128921088, 6135478762187776, 315609465774936064, 17515027337549545472, 1043104219010147483648, 66358462250378681614336, 4491141928841064201846784, 322219449242531127348887552
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Links

Crossrefs

Cf. A000108, A195727, A295237, A295254, A295255, A295256, A295258.

Programs

  • Maple
    S:= series(cot(x)*(1 - sqrt(1 - 4*tan(x)))/2, x, 32):
seq(n!*coeff(S,x,n),n=0..30); # Robert Israel, Nov 18 2017
  • Mathematica
    nmax = 19; CoefficientList[Series[Cot[x] (1 - Sqrt[1 - 4 Tan[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Tan[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - tan(x)/(1 - tan(x)/(1 - tan(x)/(1 - tan(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(17/2) * n^(n-1) / (exp(n) * (arctan(1/4))^(n-1/2)). - Vaclav Kotesovec, Nov 18 2017

A295237 Expansion of e.g.f. csc(x)*(1 - sqrt(1 - 4*sin(x)))/2.

Original entry on oeis.org

1, 1, 4, 29, 320, 4741, 88384, 1988489, 52448000, 1587545161, 54252120064, 2066298252149, 86799115489280, 3986897970744781, 198795278022098944, 10694247962623751009, 617392620634705756160, 38074395493710549747601, 2498063366053169206657024, 173745719989547715852773069
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A195621, A295254, A295255, A295256, A295257, A295258.

Programs

  • Maple
    a:=series(csc(x)*(1-sqrt(1-4*sin(x)))/2,x=0,20): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Csc[x] (1 - Sqrt[1 - 4 Sin[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Sin[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - sin(x)/(1 - sin(x)/(1 - sin(x)/(1 - sin(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2) * 15^(1/4) * n^(n-1) / (exp(n) * (arcsin(1/4))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017

A295258 Expansion of e.g.f. coth(x)*(1 - sqrt(1 - 4*tanh(x)))/2.

Original entry on oeis.org

1, 1, 4, 28, 304, 4456, 82144, 1827568, 47674624, 1427337856, 48248157184, 1817752215808, 75534405842944, 3432099993158656, 169290181445914624, 9009094978010165248, 514518446264601739264, 31389459744670699257856, 2037360033664565682110464, 140182487701223036287909888
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A295237, A295254, A295255, A295256, A295257.

Programs

  • Maple
    a:=series(coth(x)*(1-sqrt(1-4*tanh(x)))/2,x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Coth[x] (1 - Sqrt[1 - 4 Tanh[x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-Tanh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - tanh(x)/(1 - tanh(x)/(1 - tanh(x)/(1 - tanh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(15) * 2^(n-1) * n^(n-1) / (exp(n) * (log(5/3))^(n - 1/2)). - Vaclav Kotesovec, Nov 18 2017
Showing 1-6 of 6 results.

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