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.

A096050 Decimal expansion of lim_{n->oo} B(2n,7)/(B(2n)*49^n) (see comment for B(n,k) definition).

Original entry on oeis.org

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

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Author

Benoit Cloitre, Jun 17 2004

Keywords

Comments

B(n,p) = Sum_{i=0..n} p^i * Sum_{j=0..i} binomial(n,j)*B(j) where B(k) = k-th Bernoulli number. B(2n,p)/B(2n) takes integer values for all n if p=1,2,3,4,6. p=5 is the smallest integer for which B(2n,5)/B(2n) is not always integer-valued. And lim_{n->oo} B(2n,5)/(B(2n)*25^n) = (21-sqrt(5))/16.

Crossrefs

Cf. A096045, A096046, A096047, A096048, A096049.

Programs

  • Mathematica
    RealDigits[x/.FindRoot[ 1728x^3-6192x^2+7368x-2911==0,{x,1}, WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Feb 19 2012 *)
  • PARI
    solve(q=1,1.1,1728*q^3-6192*q^2+7368*q-2911)

Formula

Limit_{n->oo} B(2n, 7)/(B(2n)*49^n) = 1.0627516996902110782... is the smallest root of 1728*X^3 - 6192*X^2 + 7368*X - 2911 = 0.

A096051 Decimal expansion of lim_{n->oo} B(2n,8)/(B(2n)*64^n) (see comment for B(n,k) definition).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jun 17 2004

Keywords

Comments

B(n,p) = Sum_{i=0..n} (p^i * Sum_{j=0..i} binomial(n,j)*B(j)) where B(k) is the k-th Bernoulli number.

Examples

			1.04184188840192178222845080541359299438788058033021...

Crossrefs

Cf. A027641, A027642, A096045, A096046, A096047, A096048, A096049, A096050.

Programs

  • Mathematica
    RealDigits[(16 - Sqrt[2])/14, 10, 100][[1]] (* Amiram Eldar, May 08 2022 *)
  • PARI
    (16-sqrt(2))/14

Formula

Equals (16-sqrt(2))/14.

A096052 Decimal expansion of lim_{n->oo} B(2n,5)/(B(2n)*25^n) (see comment for B(n,k) definition).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jun 17 2004

Keywords

Comments

B(n,p) = Sum_{i=0..n} (p^i * Sum_{j=0..i} binomial(n,j)*B(j)) where B(k) is the k-th Bernoulli number.

Examples

			1.17274575140626314397442664570429523528496135252427...

Crossrefs

Cf. A027641, A027642, A096045, A096046, A096047, A096048, A096049, A096050, A096051.

Programs

  • Mathematica
    RealDigits[(21 - Sqrt[5])/16, 10, 100][[1]] (* Amiram Eldar, May 08 2022 *)
  • PARI
    (21-sqrt(5))/16

Formula

Equals (21-sqrt(5))/16.
Showing 1-3 of 3 results.

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