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A282496
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'Somos expansion' of Pi: Pi=a(0)*sqrt(a(1)*sqrt(a(2)*sqrt(a(3)*sqrt(...)))). a(n)=floor(x(n)), x(n)=x(n-1)^2/a(n-1)^2, x(0)=Pi.
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1
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3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1
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OFFSET
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0,1
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COMMENTS
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1<=a(n)<=3 for all n. Reasoning: for x>1 it follows that 1<x/floor(x)<2.
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LINKS
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FORMULA
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Product_{k>=0} a(k)^(1/2^k) = Pi.
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EXAMPLE
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Integer part of Pi is 3. Integer part of Pi^2/9 is 1.
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MATHEMATICA
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$MaxExtraPrecision = 1000;
x00 = Pi;
x0 = x00;
Nm = 130;
j = 1;
Res = Table[1, {j, 1, Nm}];
While[j < Nm, Res[[j]] = Floor[x0]; x0 = N[(x0/ Res[[j]])^2, 20000];
j++];
Res
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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