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 1-4 of 4 results.

A129404 Decimal expansion of L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

Original entry on oeis.org

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

Views

Author

Stuart Clary, Apr 15 2007

Keywords

Comments

Contributed to OEIS on Apr 15 2007 -- the 300th anniversary of the birth of Leonhard Euler.

Examples

			L(3, chi3) = 0.8840238117500798567430579168710118077...

References

  • Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292.

Links

Crossrefs

Cf. A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665.
Cf. A344778, A344727, A344688.

Programs

  • Mathematica
    nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 10^(-nmax), 10, nmax] ]
  • PARI
    4*Pi^3/81/sqrt(3) \\ Charles R Greathouse IV, Sep 02 2024

Formula

chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A102283 (A049347 shifted).
Series: L(3, chi3) = Sum_{k>=1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(3*k+1)^3 - 1/(3*k-1)^3 ). - Sean A. Irvine, Aug 17 2021
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^3)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^3)^(-1). - Amiram Eldar, Nov 06 2023

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009

A344688 Decimal expansion of 3236 * Pi^9 / (55801305 * sqrt(3)).

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Aug 17 2021

Keywords

Examples

			0.998050195657077237227863822730...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (310).

Links

Crossrefs

Cf. A129404, A344778, A344727.

Programs

  • Mathematica
    RealDigits[3236 * Pi^9 / (55801305 * Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

Formula

Equals 2^2 * 809 * Pi^9 / (3^13 * 5 * 7 * sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(3*k+1)^9 - 1/(3*k-1)^9 ).
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^9)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^9)^(-1). - Amiram Eldar, Nov 06 2023

A344727 Decimal expansion of 56 * Pi^7 / (98415 * sqrt(3)).

Original entry on oeis.org

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

Views

Author

Sean A. Irvine, Aug 17 2021

Keywords

Examples

			0.9922365295225111693516317453513060...

References

  • L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (310).

Links

Crossrefs

Cf. A129404, A344778, A344688.

Programs

  • Mathematica
    RealDigits[56 * Pi^7 / (98415 * Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

Formula

Equals 2^3 * 7 * Pi^7 / (3^9 * 5 * sqrt(3)).
Equals 1 + Sum_{k>=1} ( 1/(3*k+1)^7 - 1/(3*k-1)^7 ).
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^7)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^7)^(-1). - Amiram Eldar, Nov 06 2023

A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 15, 81, 241, 624, 1215, 2402, 3855, 6561, 9360, 14640, 19521, 28562, 36030, 50544, 61681, 83520, 98415, 130322, 150384, 194562, 219600, 279840, 312255, 390001, 428430, 531441, 578882, 707280, 758160, 923522, 986895, 1185840, 1252800, 1498848, 1581201
Offset: 1

Views

Author

Michael Somos, Dec 26 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Convolution of A008653 and A106402.

Examples

			G.f. = x + 15*x^2 + 81*x^3 + 241*x^4 + 624*x^5 + 1215*x^6 + 2402*x^7 + ...

Links

Crossrefs

Cf. A004016, A005882, A005928, A008653, A106402, A344778.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(3), 5),37);  A[2];
  • Mathematica
    a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (With[{s = {1, -1, 0}[[Mod[#, 3, 1]]]}, ((#^4)^(#2 + 1) - s^(#2 + 1)) / (#^4 - s)] & @@@ FactorInteger[n])];
  • PARI
    {a(n) = my(A, U1, u3, U9); if( n<1, 0, n--; A = x * O(x^n); U1 = eta(x + A)^3; u3 = eta(x^3 + A); U9 = eta(x^9 + A)^3; polcoeff( U1 * u3^7 * (1 + 9*x*U9/U1)^2, n))};
  • PARI
    {a(n) = my(A, p, e, s); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, p^(4*e), s=-(-1)^(p%3);  ((p^4)^(e+1) - s^(e+1)) / (p^4 - s))))};

Formula

a(n) is multiplicative with a(p^e) = ((p^4)^(e+1) - s^(e+1)) / (p^4 - s) where s = 0 if p = 3, s = 1 if p == 1 (mod 3), s = -1 if p == 2 (mod 3).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 4*Pi^5/(729*sqrt(3)) = 0.9694405... (A344778). - Amiram Eldar, Nov 09 2023
Showing 1-4 of 4 results.

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