cp's OEIS Frontend

A100338 Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).

%I A100338 #12 Apr 25 2022 16:58:33
%S A100338 1,3,5,3,8,7,1,1,2,8,4,2,9,8,8,2,3,7,4,3,8,8,8,9,4,0,8,4,0,1,6,6,0,8,
%T A100338 1,2,4,2,2,7,3,3,3,4,1,6,8,1,2,1,1,8,5,5,6,9,2,3,6,7,2,6,4,9,7,8,7,0,
%U A100338 0,1,8,4,2,0,6,4,8,2,6,0,5,4,8,4,3,1,9,6,9,7,6,0,1,7,4,6,5,6,9,7,9,6,6,8,5
%N A100338 Decimal expansion of the constant x whose continued fraction expansion equals A006519 (highest power of 2 dividing n).
%C A100338 This constant x has the special property that the continued fraction expansion of 2*x results in the continued fraction expansion of x interleaved with 2's: contfrac(x) = [1;2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,...A006519(n),... ] while contfrac(2*x) = [2;1, 2,2, 2,1, 2,4, 2,1, 2,2, 2,1, 2,8,... 2, A006519(n),...].
%C A100338 The continued fraction of x^2 has large partial quotients (see A100864, A100865) that appear to be doubly exponential.
%H A100338 Dzmitry Badziahin and Jeffrey Shallit, <a href="http://arxiv.org/abs/1505.00667">An Unusual Continued Fraction</a>, arXiv:1505.00667 [math.NT], 2015.
%H A100338 Dzmitry Badziahin and Jeffrey Shallit, <a href="https://doi.org/10.1090/proc/12848">An unusual continued fraction</a>, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.
%e A100338 1.353871128429882374388894084016608124227333416812118556923672649787001842...
%t A100338 cf = ContinuedFraction[ Table[ 2^IntegerExponent[n, 2], {n, 1, 200}]]; RealDigits[ FromContinuedFraction[cf // Flatten] , 10, 105] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%o A100338 (PARI) /* This PARI code generates 1000 digits of x very quickly: */ {x=sqrt(2);y=x;L=2^10;for(i=1,10,v=contfrac(x,2*L); if(2*L>#v,v=concat(v,vector(2*L-#v+1,j,1))); if(2*L>#w,w=concat(w,vector(2*L-#w+1,j,1))); w=vector(2*L,n,if(n%2==1,2,w[n]=v[n\2]));w[1]=floor(2*x); CFW=contfracpnqn(w);x=CFW[1,1]/CFW[2,1]*1.0/2;);x}
%o A100338 (PARI) {CFM=contfracpnqn(vector(1500,n,2^valuation(n,2))); x=CFM[1,1]/CFM[2,1]*1.0}
%Y A100338 Cf. A006519, A100863, A100864, A100865.
%K A100338 nonn,cons
%O A100338 1,2
%A A100338 _Paul D. Hanna_, Nov 17 2004