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A219014
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Numerators in a product expansion for sqrt(2).
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3
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OFFSET
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0,1
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COMMENTS
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a(3) has 96 digits and a(4) has 479 digits.
Iterating the algebraic identity sqrt(1 + 4/x) = (1 + 2*(x + 2)/(x^2 + 3*x + 1)) * sqrt(1 + 4/(x*(x^2 + 5*x + 5)^2)) produces a rapidly converging product expansion sqrt(1 + 4/x) = product {n = 0..inf} (1 + 2*a(n)/b(n)), where a(n) and b(n) are integer sequences when x is a positive integer.
The present case is when x = 4. The denominators b(n) are in A219015. See also A219010 (x = 1) and A219012 (x = 2).
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LINKS
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FORMULA
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Let tau = 3 + 2*sqrt(2). Then a(n) = tau^(5^n) + 1/tau^(5^n).
Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 6.
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EXAMPLE
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The first two terms of the product give 18 correct decimal places of sqrt(2): (1 + 2*6/29)*(1 + 2*6726/45232349) = 1.41421 35623 73095 048(5...).
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CROSSREFS
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KEYWORD
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nonn,easy,bref
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AUTHOR
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STATUS
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approved
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