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.

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

Author

Clark Kimberling, Dec 06 2015

Keywords

• nonn
• cons

Comments

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.

Examples

			1.195955786017513596003474800021...
		

Crossrefs

Cf. A000045, A001622, A290565.

Cf. A265288, A265289.

Programs

Maple

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

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, 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}]

Formula

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