A265289 -id:A265289 - cp's OEIS frontend

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.

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

Clark Kimberling, Dec 06 2015

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

			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) + ...

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

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

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 *)

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)

A265290 Decimal expansion of Sum_{n>=1} |phi - c(n)|, where phi is the golden ratio (A001622) and c(n) are the convergents to phi.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 06 2015

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

			1.195955786017513596003474800021...

Cf. A000045, A001622, A290565.

Cf. A265288, A265289.

x := (3 - sqrt(5))/2:
evalf(sqrt(5)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..16), 100); # Peter Bala, Aug 21 2022

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, dev(x) *)
d[x_] := If[IntegerQ[1000!*x], Total[Abs[x - Convergents[x]]],
  Total[Abs[x - Convergents[x, 30]]]]
Plot[{d[x], 1.195}, {x, 0, 1}]

Equals Sum_{n>=1} 1/(F(2*n-1)*F(2*n)), where F(n) is the n-th Fibonacci number (A000045).
From Amiram Eldar, Oct 05 2020: (Start)
Equals Sum_{k>=1} 1/(phi^k * F(k)).
Equals sqrt(5) * Sum_{k>=1} 1/(phi^(2*k) - (-1)^k) = sqrt(5) * Sum_{k>=1} (-1)^(k+1)/(phi^(2*k) + (-1)^k).
Equals (A290565 + 1/phi)/2. (End)
A rapidly converging series for the constant is sqrt(5)*Sum_{k >= 1} x^(k^2)*(1 + x^(2*k))/(1 - x^(2*k)), where x = (3 - sqrt(5))/2. See A112329. - Peter Bala, Aug 21 2022

cp's OEIS Frontend

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.

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