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 1-8 of 8 results.

A230151 Decimal expansion of the positive real solution of the equation x^4 + x^3 - 1 = 0.

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Oct 11 2013

Keywords

Comments

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-3.

Examples

			0.8191725133961644396995711883424270403484978325537129656...

Links

Crossrefs

Cf. A086106 (other real root).
Cf. A075778, A230152-A230158.

Programs

  • Maple
    with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-3);
  • Mathematica
    Root[(#^4+#^3-1)&, 2] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)
  • PARI
    polrootsreal(x^4+x^3-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

A230154 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=6.

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Oct 11 2013

Keywords

Comments

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-6.

Examples

			0.8986537126286992932608757220468058862604482200934396966...

Links

Crossrefs

Cf. A075778, A230151, A230152, A230153, A230155, A230156, A230157, A230158, A230160, A377081.

Programs

  • Maple
    with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-6);
  • Mathematica
    RealDigits[x/.FindRoot[x^7+x^6==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 30 2013 *)

Formula

Equals 1/A230160. - Hugo Pfoertner, Oct 15 2024

A230153 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=5.

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Oct 11 2013

Keywords

Comments

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-5.

Examples

			0.8812714616335695944076491628413720252791939793788952636...

Links

Crossrefs

Cf. A075778, A230151, A230152, A230154-A230158.

Programs

  • Maple
    with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-5);
  • Mathematica
    Root[x^6 + x^5 - 1, 2] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A358939 Decimal expansion of the real root of x^5 + x^3 - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Dec 15 2022

Keywords

Comments

The other (complex) roots are 0.217853219392291296... + 1.16695124566484991...*i and -0.636663106805772389... + 0.664701565064356279...*i, together with their conjugates.

Examples

			0.83761977482696218499752729419180609392505451858960237912530556912378529...

Crossrefs

Cf. A160155, A230152, A358940, A358941, A358942.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^5 + x^3 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

A358940 Decimal expansion of the real root of x^5 - x^3 - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Dec 12 2022

Keywords

Comments

The other (complex) roots are 0.340794866197006415... + 0.785423103049449080...*i and -0.959047717892755927... + 0.428365956254189316...*i, and their conjugates.

Examples

			1.23650570339149902433757480097614678268104294354611496776617384170726143...

Crossrefs

Cf. A160155, A230152, A358939, A358941, A358942.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^5 - x^3 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

A358941 Decimal expansion of the real root of x^5 + x^2 - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Dec 15 2022

Keywords

Comments

The other (complex) roots are 0.464912201602897854... + 1.07147384027026940...*i, and -0.869277501842593861 + 0.388269406599740355...*i, and their conjugates.

Crossrefs

Cf. A160155, A230152, A358939, A358940, A358942.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^5 + x^2 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)
  • PARI
    solve(x=0, 1, x^5 + x^2 - 1) \\ Michel Marcus, Dec 19 2022

Formula

0.808730600479392013738554526511400064951377351559313075548116401836543340...

A358942 Decimal expansion of the real root of x^5 - x^2 - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Dec 15 2022

Keywords

Comments

The other (complex) roots are 0.154589676718332223... + 0.828074133201299911...*i, and -0.751519232378943729... + 0.784615921039447991...*i, together with their conjugates.

Examples

			1.19385911132122301200902074629803112451452426948644450960208140159603556...

Crossrefs

Cf. A160155, A230152, A358939, A358940, A358941.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^5 - x^2 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

A377080 G.f.: Sum_{k>=1} x^(2*k^2) * Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 15 2024

Keywords

Links

Crossrefs

Cf. A306734, A350890, A350893, A377081.
Cf. A230152.

Programs

  • Mathematica
    nmax = 150; CoefficientList[Series[Sum[x^(2*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/2]}], {x, 0, nmax}], x]

Formula

a(n) ~ (1+r) * exp(sqrt((40*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((4 + 5*r)*n)), where r = A230152 = 0.856674883854502874852324... is the real root of the equation r^4*(1+r) = 1.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.