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A265288
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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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24
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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
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OFFSET
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0,1
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COMMENTS
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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)|
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LINKS
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FORMULA
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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
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)
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EXAMPLE
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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) + ...
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MAPLE
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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
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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 *)
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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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