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A282496 '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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
1<=a(n)<=3 for all n. Reasoning: for x>1 it follows that 1<x/floor(x)<2.
LINKS
FORMULA
Product_{k>=0} a(k)^(1/2^k) = Pi.
EXAMPLE
Integer part of Pi is 3. Integer part of Pi^2/9 is 1.
MATHEMATICA
$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
CROSSREFS
Cf. A000796 (digits), A100044 (Pi^2/9), A001203 (continued fraction), A276459 (another nested radical expansion).
Sequence in context: A344759 A337820 A322127 * A253238 A249773 A030369
KEYWORD
nonn
AUTHOR
Yuriy Sibirmovsky, Feb 16 2017
STATUS
approved

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)