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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.

%I A276459 #31 Feb 16 2025 08:33:36
%S A276459 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 A276459 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 A276459 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 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.
%C A276459 Similar to Bolyai expansion. Uses the fact that for 0<p<1, 2<(2+p)^2-2<7.
%H A276459 Yuriy Sibirmovsky, <a href="/A276459/b276459.txt">Table of n, a(n) for n = 1..250</a>
%H A276459 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BolyaiExpansion.html">Bolyai Expansion</a>.
%e A276459 Pi^2=7+2+p1, thus a(1)=7;
%e A276459 (2+p1)^2=6+2+p2, thus a(2)=6;
%e A276459 (2+p2)^2=2+2+p3, thus a(3)=2; ... 0<pn<1.
%t A276459 Nm=100;
%t A276459 A=Table[1,{j,1,Nm}];
%t A276459 V=Table[1,{j,1,Nm}];
%t A276459 P=Pi;
%t A276459 p0=P;
%t A276459 Do[p1=Floor[p0^2]-2;
%t A276459 A[[j]]=p1;
%t A276459 p0=N[2+p0^2-Floor[p0^2],300],{j,1,Nm}];
%t A276459 Do[v0=Sqrt[A[[n]]];
%t A276459 Do[v1=A[[n-j]]+v0;
%t A276459 v0=Sqrt[v1],{j,1,n-1}];
%t A276459 V[[n]]=v0,{n,1,Nm}];
%t A276459 A
%Y A276459 Cf. A000796 (digits), A001203 (continued fraction).
%K A276459 nonn
%O A276459 1,1
%A A276459 _Yuriy Sibirmovsky_, Sep 03 2016