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A243261
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Decimal expansion of a Shapiro-Drinfeld constant, known as Gauchman's constant, related to the difference of cyclic sums (negated).
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8
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0, 2, 1, 9, 8, 7, 5, 2, 1, 8, 1, 3, 3, 5, 3, 7, 7, 9, 7, 9, 8, 1, 7, 2, 0, 4, 0, 0, 2, 1, 3, 1, 7, 3, 1, 9, 0, 6, 7, 4, 6, 1, 3, 6, 4, 6, 5, 4, 0, 8, 5, 8, 1, 9, 0, 5, 0, 4, 6, 9, 5, 7, 9, 1, 6, 5, 4, 0, 0, 5, 5, 0, 7, 4, 1, 9, 2, 8, 9, 3, 0, 2, 4, 9, 9, 4, 4, 3, 3, 1, 0, 1, 4, 7, 9, 6, 0, 7, 3, 4
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OFFSET
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0,2
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COMMENTS
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Let a_i > 0 for i = 1, ..., n, and define the cyclic sums E_n = a_1/(a_2 + a_3) + a_2/(a_3 + a_4) + ... + a_n/(a_1 + a_2) and F_n = a_1/(a_1 + a_2) + a_2/(a_2 + a_3) + ... + a_n/(a_n + a_1). Gauchman (1998) proved that E_n - F_n >= lambda*n for all n >= 1, where lambda = -0.02198... (this constant) is the best constant. His proof was not published, however, as a solution to Problem 10528(b) in the American Mathematical Monthly (see the link below). Only a comment was made on p. 474. - Petros Hadjicostas, Jun 02 2020
Named after the mathematicians Harold Seymour Shapiro (1928-2021) and Vladimir Drinfeld (b. 1954). Alternatively, named after the American mathematician Hillel V. Gauchman (1937-2016). - Amiram Eldar, May 29 2021
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 210.
Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.
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LINKS
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Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, Solution to Problem 10528(a,b), American Mathematical Monthly, 105 (1998), 473-474. [A comment was made about Hillel Gauchman's solution to part (b) of the problem that involves this constant, but no solution was published.]
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FORMULA
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We solve the following system of equations:
exp(-c) = (exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2 and
2*(1 - exp(b/2)) = (exp(b) + exp(b/2))*(exp(-c)*(1 + c - b) - 1).
Then the constant equals (exp(-c)*(1 + c) - 1)/2.
It turns out that b = -A335810 = -0.387552... and c = A335809 = 0.330604... even though A335810 and A335809 are also involved in the calculation of the Shapiro cyclic sum constant mu = A086278.
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EXAMPLE
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-0.02198752181335377979817204... = -1 + 0.9780124781866462202018...
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MATHEMATICA
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eq = E^(u/2) + 2*E^u + E^(3*u/2) + E^(u + v) == E^v + 2*E^(u/2 + v) && 2*(y + E^(u/2 - v) + 1) == (u - 2)/E^v + 4/E^(u/2) && u + 6*E^(u/2) + 4*E^u + 4*E^(u/2 + v) + 1 == v + 9*E^v; y0 = y /. FindRoot[eq , {{y, 0}, {u, -1/3}, {v, 1/3}}, WorkingPrecision -> 105]; RealDigits[y0, 10, 99] // First
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PROG
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(PARI) default("realprecision", 200)
c(b) = -log((exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2);
a = solve(b=-2, 0, (exp(b) + exp(b/2))*(-1 + exp(-c(b))*(1 + c(b) - b)) - 2*(1 - exp(b/2)));
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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