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A282089
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Decimal expansion of constant 1.287194... related to a conjectural Viète-like formula for Pi.
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1
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1, 2, 8, 7, 1, 9, 4, 0, 3, 6, 0, 6, 7, 9, 2, 4, 0, 1, 7, 0, 2, 0, 9, 2, 7, 8, 0, 7, 5, 8, 1, 1, 9, 8, 7, 6, 4, 4, 0, 8, 3, 5, 4, 3, 5, 6, 6, 9, 9, 2, 7, 8, 0, 5, 4, 4, 8, 6, 1, 4, 1, 2, 9, 3, 2, 7, 1, 4, 5, 2, 8, 3, 9, 1, 4, 4, 8, 7, 2, 0, 2, 2, 1, 1, 2, 3, 7, 9, 0, 7, 9, 9, 2, 6, 0, 9, 3, 4, 0, 3, 3, 9, 9, 8
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OFFSET
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1,2
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COMMENTS
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Conjecture: Pi = lim_{k -> infinity} 2^{k + 1}*(1 - c_k), where the variable c_k is defined by a set of the Viète-like recurrence relations {a_1 = sqrt(2), a_k = sqrt(2 + a_{k - 1}), b_k = sqrt(2 - a_k)/a_{k + 1}, c_1 = b_1, c_k = (c_{k - 1} + b_k)/(1 - c_{k - 1}*b_k)}. From this conjecture it follows that Sum_{k >= 1} (1 - c_k) is convergent [Abrarov and Quine].
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LINKS
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FORMULA
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Sum_{k >= 1} (1 - c_k) = 1.287194... , where c_k is computed by the recurrence equations a_1 = sqrt(2), a_k = sqrt(2 + a_{k - 1}), b_k = sqrt(2 - a_k)/a_{k + 1}, c_1 = b_1 and c_k = (c_{k - 1} + b_k)/(1 - c_{k - 1}*b_k).
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EXAMPLE
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1.287194036067924017020927807581...
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MATHEMATICA
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Clear[a, b, c]
a[k_] := N[Nest[Sqrt[2 + #1] &, 0, k], 100]
b[k_] := b[k] = Sqrt[2 - a[k]]/a[k + 1]
c[1] := b[1] = b[1]
c[k_] := c[k] = (c[k - 1] + b[k])/(1 - c[k - 1]*b[k])
k := 90
Print["Index k = ", k]
m := 1
Print["Power m = ", m]
(* The equation (12) *)
apprPi := 2^(k + 1)*(1 - c[k]^m)
Print["Actual value of Pi is ", N[Pi, 30]]
Print["At k = ", k, " the approximated value of Pi is ", N[apprPi, 30]]
K := 300
Print["Truncating integer K = ", K]
Print["Computing the digits ..."]
RealDigits[N[Sum[1 - c[k]^m, {k, 1, K}], 30]][[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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