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

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

Author

Clark Kimberling, Dec 06 2015

Keywords

• nonn
• cons

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