A282496 - OEIS

%I #10 Feb 19 2017 14:04:48

%S 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,

%T 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,

%U 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

%N '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.

%C 1<=a(n)<=3 for all n. Reasoning: for x>1 it follows that 1<x/floor(x)<2.

%H Yuriy Sibirmovsky, <a href="/A282496/b282496.txt">Table of n, a(n) for n = 0..1999</a>

%F Product_{k>=0} a(k)^(1/2^k) = Pi.

%e Integer part of Pi is 3. Integer part of Pi^2/9 is 1.

%t $MaxExtraPrecision = 1000;

%t x00 = Pi;

%t x0 = x00;

%t Nm = 130;

%t j = 1;

%t Res = Table[1, {j, 1, Nm}];

%t While[j < Nm, Res[[j]] = Floor[x0]; x0 = N[(x0/ Res[[j]])^2, 20000];

%t   j++];

%t Res

%Y Cf. A000796 (digits), A100044 (Pi^2/9), A001203 (continued fraction), A276459 (another nested radical expansion).

%K nonn

%O 0,1

%A _Yuriy Sibirmovsky_, Feb 16 2017