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 1-3 of 3 results.

A173987 a(n) = denominator of ((Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9), where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

1, 4, 100, 1600, 193600, 9486400, 2741569600, 2741569600, 1450290318400, 245099063809600, 206128312663873600, 3298053002621977600, 3298053002621977600, 1190597133946533913600, 2001393782164123508761600
Offset: 0

Views

Author

Artur Jasinski, Mar 04 2010

Keywords

Links

Crossrefs

For numerators see A173985.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173986.

Programs

  • Magma
    [1] cat [Denominator((&+[9/(3*k+2)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
  • Maple
    a := n -> (Zeta(0,2,2/3) - Zeta(0,2,n+2/3))/9:
seq(denom(a(n)), n=0..14); # Peter Luschny, Nov 14 2017
  • Mathematica
    Table[FunctionExpand[(1/9)*(4*(Pi^2)/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])], {n, 0, 20}] // Denominator (* Vaclav Kotesovec, Nov 13 2017 *)
Denominator[Table[Sum[9/(3*k + 2)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)
  • PARI
    for(n=0,20, print1(denominator(9*sum(k=0,n-1, 1/(3*k+2)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

Formula

a(n) = denominator of 2*(Pi^2)/3 - J - Zeta(2,(3*n+2)/3), where Zeta is the Hurwitz Zeta function and J is the constant A173973.
a(n) = denominator of Sum_{k=0..(n-1)} 9/(3*k+2)^2. - G. C. Greubel, Aug 23 2018

Extensions

Name simplified by Peter Luschny, Nov 14 2017

A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

1, 1, 16, 784, 19600, 3312400, 52998400, 19132422400, 2315023110400, 57875577760000, 57875577760000, 55618430227360000, 16073726335707040000, 22004931353582937760000, 22004931353582937760000
Offset: 0

Views

Author

Artur Jasinski, Mar 04 2010

Keywords

Links

Crossrefs

For the numerators see A173982.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173983, A173985, A173986, A173987.

Programs

  • Magma
    [1,1] cat [Denominator((&+[9/(3*k+1)^2: k in [1..n-1]])): n in [2..20]]; // G. C. Greubel, Aug 24 2018
  • Maple
    a := n -> Zeta(0,2,1/3) - Zeta(0,2,n+1/3):
seq(denom(a(n)), n=0..14); # Peter Luschny, Nov 14 2017
  • Mathematica
    Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {n, 0, 20}] // Denominator (* Vaclav Kotesovec, Nov 13 2017 *)
Denominator[Table[Sum[9/(3*k + 1)^2, {k, 1, n - 1}], {n, 0, 30}]] (* G. C. Greubel, Aug 24 2018 *)
  • PARI
    for(n=0,20, print1(denominator(sum(k=1,n-1, 9/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 24 2018

Formula

a(n) = denominator of 2*(Pi^2)/3 + J - Zeta(2,(3*n+1)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.
a(n) = denominator of Sum_{k=1..(n-1)} 9/(3*k+1)^2. - G. C. Greubel, Aug 24 2018

Extensions

Name simplified by Peter Luschny, Nov 14 2017

A173986 a(n) = numerator((Psi(1, 2/3) - Psi(1, n+2/3))/9), where Psi(1, z) is the Trigamma function.

Original entry on oeis.org

0, 1, 29, 489, 60769, 3026081, 884023809, 890877733, 474015890357, 80471258049933, 67921427083803253, 1089963588226225073, 1092655876391630769, 395273284628034202009, 665644988593672027490729
Offset: 0

Views

Author

Artur Jasinski, Mar 04 2010

Keywords

Comments

a(n+1)/A173987(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(3*k+2)^2. The limit n -> infinity is given in A294967 as the Hurwitz Zeta function or the Trigamma function (1/9)*Zeta(2, 2/3) = (1/9)*Psi(1, 2/3) = 0.3404306010 ... - Wolfdieter Lang, Nov 12 2017

Examples

			The rationals a(n)/A173987(n) begin 0/1, 1/4, 29/100, 489/1600, 60769/193600, 3026081/9486400, 884023809/2741569600, 890877733/2741569600, ... - _Wolfdieter Lang_, Nov 12 2017

Links

Crossrefs

For denominators see A173987.
For 9*A173986 see A173985.
Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173985, A294967.

Programs

  • Magma
    [0] cat [Numerator((&+[2/(3*k+2)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018
  • Maple
    r := n -> (Psi(1, 2/3) - Psi(1, n+2/3))/9:
seq(numer(simplify(r(n))), n=0..14); # Peter Luschny, Nov 13 2017
  • Mathematica
    Table[Numerator[FunctionExpand[(4*Pi^2/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])/9]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[2/(3*k + 2)^2, {k, 0, n - 2}], {n, 1, 20}]] (* G. C. Greubel, Aug 23 2018 *)
  • PARI
    for(n=1,20, print1(numerator(sum(k=0,n-2, 2/(3*k+2)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

Formula

a(n) = numerator(r(n)) with r(n) = (1/9)*(4*(Pi^2)/3 - Zeta(2, 1/3) - Zeta(2, (3*n+2)/3)) = (1/9)*(Zeta(2, 2/3) - Zeta(2, (3*n+2)/3)) with the Hurwitz Zeta function Zeta(2, q). This becomes the formula given in the name. - Wolfdieter Lang, Nov 13 2017
a(n) = numerator of (1/9)*(2(Pi^2)/3 - J - Zeta(2, (3n+2)/3)) where J is the constant A173973 [which becomes the preceding formula].
a(n) = numerator of Sum_{k=0..(n-2)} 2/(3*k+2)^2. - G. C. Greubel, Aug 23 2018

Extensions

Name simplified by Peter Luschny, Nov 13 2017
Showing 1-3 of 3 results.

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