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.

A258414 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(24*k)) dx.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 29 2015

Keywords

Comments

Integral_{x=-1..1} Product_{k>=1} (1-x^(24*k)) dx = Pi^2/(3*sqrt(3)) = 1.89940625258801878... . - Vaclav Kotesovec, Jun 02 2015
Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=2. - Jianing Song, Nov 16 2019

Examples

			0.9497031262940093952634984917457415158736519509096929448809176543683...

Links

Crossrefs

Cf. A258232, A258406, A258408.
Cf. A196530, A110161.
Cf. A346451, A346450, A346449.

Programs

  • Maple
    evalf(Pi^2/(6*sqrt(3)), 120);
  • Mathematica
    RealDigits[Pi^2/(6*Sqrt[3]),10,120][[1]]
N[Sum[(-1)^n/(12*n*(3n-1)+1),{n,-Infinity,Infinity}],105]

Formula

Equals Pi^2/(6*sqrt(3)).
Equals Sum_{k>=1} A110161(n)/k^2 = Sum_{k>=1} Kronecker(12,k)/k^2. - Jianing Song, Nov 16 2019
Equals -Integral_{x=0..oo} log(x)/(x^6 + 1) dx. - Amiram Eldar, Aug 12 2020
Equals 1 + Sum_{k>=1} ( (-1)^k/(6*k-1)^2 + (-1)^k/(6*k+1)^2 ). - Sean A. Irvine, Jul 18 2021
Equals 1/(Product_{p prime == 1 or 11 (mod 12)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 12)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A346449 Decimal expansion of 257543 * Pi^8 / (1410877440 * sqrt(3)).

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Jul 18 2021

Keywords

Comments

Note Shamos has a typo, 25743 instead of 257543.

Examples

			0.9999972722389309471467818749637839272444862497...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (329).

Links

Crossrefs

Cf. A258414, A346451, A346450.

Programs

  • Mathematica
    First[RealDigits[N[11*13*1801*Pi^8/(2^11*3^9*5*7*Sqrt[3]),100]]] (* Stefano Spezia, Jul 19 2021 *)

Formula

Equals 11 * 13 * 1801 * Pi^8 / (2^11 * 3^9 * 5 * 7 * sqrt(3)).
Equals 1 + Sum_{k>=1} ( (-1)^k/(6*k-1)^8 + (-1)^k/(6*k+1)^8 ).

A346451 Decimal expansion of 23 * Pi^4 / (1296 * sqrt(3)).

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Jul 18 2021

Keywords

Examples

			0.99807159983792868732970969812...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (329).

Links

Crossrefs

Cf. A258414, A346450, A346449.

Programs

  • Mathematica
    First[RealDigits[N[23*Pi^4/(2^4*3^4*Sqrt[3]),100]]] (* Stefano Spezia, Jul 19 2021 *)
RealDigits[(23 Pi^4)/(1296 Sqrt[3]),10,120][[1]] (* Harvey P. Dale, Jun 20 2023 *)

Formula

Equals 23 * Pi^4 / (2^4 * 3^4 * sqrt(3)).
Equals 1 + Sum_{k>=1} ( (-1)^k/(6*k-1)^4 + (-1)^k/(6*k+1)^4 ).

A346570 Decimal expansion of 2 - 33367 * Pi^7 / 100776960.

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Jul 23 2021

Keywords

Examples

			0.9999884484387282478381315723917964998588...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (316).

Links

Crossrefs

Cf. A346573, A346572, A346571, A346449, A346450.

Programs

  • Mathematica
    RealDigits[2 - 33367 * Pi^7 / 100776960, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Formula

Equals 2 - 61 * 547 * Pi^7 / (2^10 * 3^9 * 5).
Equals 1 + Sum_{k>=1} ( ((-1)^k/(6*k-1)^7 - (-1)^k/(6*k+1)^7) ).

A346727 Decimal expansion of 361 * Pi^6 / (245760 * sqrt(2)).

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Jul 30 2021

Keywords

Comments

Shamos (2011) has incorrect formula 19^2*Pi^4 / (2^14*3*5*sqrt(3)).

Examples

			0.998573971953530547670270516106668073031954930...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (327).

Links

Crossrefs

Cf. A328895, A346728, A346260, A346450.

Programs

  • Mathematica
    RealDigits[361*Pi^6/(245760*Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Formula

Equals 19^2 * Pi^6 / (2^14 * 3 * 5 * sqrt(2)).
Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1)^6 + (-1)^k/(4*k+1)^6 ).
Showing 1-5 of 5 results.

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