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A265290
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Decimal expansion of Sum_{n>=1} |phi - c(n)|, where phi is the golden ratio (A001622) and c(n) are the convergents to phi.
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4
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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Equals Sum_{n>=1} 1/(F(2*n-1)*F(2*n)), where F(n) is the n-th Fibonacci number (A000045).
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).
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
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EXAMPLE
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1.195955786017513596003474800021...
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MAPLE
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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
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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