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 A276459 Nested radical expansion of Pi: Pi = sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + sqrt(a(4) + ...)))), with a(1) = 7 and 2 <= a(n) <= 6 for n>1. 2

%I

%S 7,6,2,6,6,5,5,2,4,6,3,4,2,4,6,3,6,3,3,5,4,3,6,3,3,3,4,3,6,6,4,3,3,4,

%T 5,5,2,6,2,5,4,3,4,6,6,2,3,5,2,3,5,4,2,3,2,4,2,6,4,6,3,3,4,3,4,6,3,4,

%U 6,5,2,2,2,3,4,5,5,5,2,4,3,6,4,3,6,3,2,6,2,4,5,6,2,3,2,5,2,3,2,3,3,5,4,4,6,4,2,4,5,4,6,5,3

%N Nested radical expansion of Pi: Pi = sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + sqrt(a(4) + ...)))), with a(1) = 7 and 2 <= a(n) <= 6 for n>1.

%C Similar to Bolyai expansion. Uses the fact that for 0<p<1, 2<(2+p)^2-2<7.

%H Yuriy Sibirmovsky, <a href="/A276459/b276459.txt">Table of n, a(n) for n = 1..250</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BolyaiExpansion.html">Bolyai Expansion</a>.

%e Pi^2=7+2+p1, thus a(1)=7;

%e (2+p1)^2=6+2+p2, thus a(2)=6;

%e (2+p2)^2=2+2+p3, thus a(3)=2; ... 0<pn<1.

%t Nm=100;

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

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

%t P=Pi;

%t p0=P;

%t Do[p1=Floor[p0^2]-2;

%t A[[j]]=p1;

%t p0=N[2+p0^2-Floor[p0^2],300],{j,1,Nm}];

%t Do[v0=Sqrt[A[[n]]];

%t Do[v1=A[[n-j]]+v0;

%t v0=Sqrt[v1],{j,1,n-1}];

%t V[[n]]=v0,{n,1,Nm}];

%t A

%Y Cf. A000796 (digits), A001203 (continued fraction).

%K nonn

%O 1,1

%A _Yuriy Sibirmovsky_, Sep 03 2016

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Last modified June 19 05:18 EDT 2021. Contains 345125 sequences. (Running on oeis4.)