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.

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A265308 Decimal expansion of Sum_{k>=1} (c(2k)-c(2k-1)), where c = convergents to e.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 13 2015

Keywords

Examples

			sum = 1.0881628951926320258095299765374284161730153...

Crossrefs

Cf. A001113, A265306, A265307, A265288 (guide).

Programs

  • Mathematica
    x = E; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265306 *)
RealDigits[s2, 10, 120][[1]]  (* A265307 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265308 *)

A265289 Decimal expansion of Sum_{n>=1} (c(2*n) - phi), where phi is the golden ratio (A001622) and c = convergents to phi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2015

Keywords

Comments

Define the upper deviance of x > 0 by dU(x) = Sum_{n>=1} (c(2*n,x) - x), where c(k,x) = k-th convergent to x. The greatest upper deviance occurs when x = golden ratio, so that this constant is the absolute maximal upper deviance.

Examples

			0.4387514109715062573556495393475271901...

Crossrefs

Cf. A000005, A000045, A001622, A265288, A265290.

Programs

  • Maple
    x := (7 - 3*sqrt(5))/2:
evalf(sqrt(5)*add(x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..12), 100); # Peter Bala, Aug 21 2022
  • Mathematica
    x = GoldenRatio; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265288 *)
RealDigits[s2, 10, 120][[1]]  (* A265289 *)
RealDigits[s1 + s2, 10, 120][[1]] (* A265290 *)

Formula

Equals Sum_{k>=1} 1/(phi^(2*k) * F(2*k)), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
Equals sqrt(5)*Sum_{k >= 1} x^(k^2)*(1 + x^k)/(1 - x^k), where x = (7 - 3*sqrt(5))/2. - Peter Bala, Aug 21 2022

A357054 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*k/Fibonacci(2*k).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Sep 10 2022

Keywords

Examples

			0.58000473950777068006747098189552280269850126096461...

Links

Crossrefs

Cf. A000045, A001519, A001906, A081068, A357053.
Cf. A079586, A153386, A153387, A158933, A265288.

Programs

  • Mathematica
    RealDigits[Sum[(-1)^(k+1)*k/Fibonacci[2*k], {k, 1, 300}], 10, 100][[1]]
  • PARI
    sumalt(k=1, (-1)^(k+1)*k/fibonacci(2*k)) \\ Michel Marcus, Sep 10 2022

Formula

Equals Sum_{k>=1} (-1)^(k+1)*k/A001906(k).
Equals (1/sqrt(5)) * Sum_{k>=1} 1/Fibonacci(2*k-1)^2 (Jennings, 1994).
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