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A265289
Decimal expansion of Sum_{n>=1} (c(2*n) - phi), where phi is the golden ratio (A001622) and c = convergents to phi.
3
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
0,1
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.
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
EXAMPLE
0.4387514109715062573556495393475271901...
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 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 06 2015
STATUS
approved