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.

A265288 Decimal expansion of Sum_{n >= 1} (phi - c(2*n-1)), where phi is the golden ratio (A001622), and c(n) is the n-th convergent to the continued fraction expansion of phi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2015

Keywords

Comments

Define the lower deviance of x > 0 by dL(x) = Sum_{n>=1} (x - c(2*n-1,x)), where c(k,x) = k-th convergent to x. The greatest lower deviance occurs when x = golden ratio, so that this constant is the absolute maximal lower deviance.
Guide to related constants (as sequences):
x Sum{x-c(2*n-1)} Sum{c(2*n)-x} Sum|c(2*n)-c(2*n-1)|
(1+sqrt(5))/2 A265288 A265289 A265290
sqrt(2) A265291 A265292 A265293
sqrt(3) A265294 A265295 A265296
sqrt(5) A265297 A265298 A265299
sqrt(6) A265300 A265301 A265302
sqrt(8) A265303 A265304 A265305
e A265306 A265307 A265308

Examples

			0.75720437504600733864782526067377483...
The convergents to x are c(1) = 1, c(2) = 2, c(3) = 3/2, c(4) = 5/3, ..., so that
A265288 = (x - 1) + (x - 3/2) + (x - 8/5) + ... ;
A265289 = (2 - x) + (5/3 - x) + (13/8 - x ) + ... ;
A265290 = (2 - 1) + (5/3 - 3/2) + (13/8 - 8/5) + ...

Crossrefs

Cf. A000045, A001227, A001622, A153386, A265289, A265290.

Programs

  • Maple
    x := -(3 - sqrt(5))/2:
evalf(sqrt(5)*add(x^(n*(n+1)/2)/(x^n - 1), n = 1..24), 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-1) * F(2*k-1)), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
From Peter Bala, Aug 19 2022: (Start)
The constant equals Sum_{k >= 1} (-1)^(k+1)/F(2*k). The constant also equals (3/5)*Sum_{k >= 1} (-1)^(k+1)/(F(2*k)*F(2*k+2)*F(2*k+4)) + 11/15.
A rapidly converging series for the constant is sqrt(5) * Sum_{k >= 1} x^(k*(k+1)/2)/ (x^k - 1) at x = phi - 2 = -(3 - sqrt(5))/2. (End)

A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

Original entry on oeis.org

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

Views

Author

Bobby Jacobs and Robert G. Wilson v, Aug 06 2017

Keywords

Comments

The constant k in A277266 such that A277266(n) ~ k*n.

Examples

			1/(1*1) + 1/(1*2) + 1/(2*3) + 1/(3*5) + ... = 1 + 1/2 + 1/6 + 1/15 + ... = 1.77387758328513234380...

Crossrefs

Cf. A000045, A000796, A001622, A001654, A002163, A002390, A079586, A094214, A265290, A277266.

Programs

  • Mathematica
    RealDigits[ Sum[1/(Fibonacci[k]*Fibonacci[k + 1]), {k, 265}], 10, 111][[1]]
  • PARI
    suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1))) \\ Michel Marcus, Feb 19 2019

Formula

Equals Sum_{n>=1} 1/(Fibonacci(n)*Fibonacci(n+1)).
Equals lim_{n->infinity} A277266(n)/n.
Equals 2 * (Sum_{k>=1} 1/(phi^k * F(k))) - 1/phi = 2 * A265290 - A094214, where phi is the golden ratio (A001622) and F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
Equals 3/2 + 10*c*Integral_{x=0..infinity} f(x) dx, where c = sqrt(5)/log(phi) = A002163/A002390, phi = (1+sqrt(5))/2 = A001622, and f(x) = sin(x)/((exp(Pi*x/(2*log(phi)))-1)*(7-2*cos(x))*(3+2*cos(x))). - Gleb Koloskov, Sep 12 2021

Extensions

More terms from Alois P. Heinz, Aug 06 2017

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
Showing 1-3 of 3 results.

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