A100338 - OEIS

A100338

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

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, 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, 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

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OFFSET

1,2

COMMENTS

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),...].

The continued fraction of x^2 has large partial quotients (see A100864, A100865) that appear to be doubly exponential.

LINKS

Table of n, a(n) for n=1..105.

Dzmitry Badziahin and Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015.

Dzmitry Badziahin and Jeffrey Shallit, An unusual continued fraction, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.

EXAMPLE

1.353871128429882374388894084016608124227333416812118556923672649787001842...

MATHEMATICA

cf = ContinuedFraction[ Table[ 2^IntegerExponent[n, 2], {n, 1, 200}]]; RealDigits[ FromContinuedFraction[cf // Flatten] , 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)

PROG

(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}

(PARI) {CFM=contfracpnqn(vector(1500, n, 2^valuation(n, 2))); x=CFM[1, 1]/CFM[2, 1]*1.0}

CROSSREFS

Cf. A006519, A100863, A100864, A100865.

Sequence in context: A110465 A354528 A328915 * A094444 A231641 A099446

Adjacent sequences: A100335 A100336 A100337 * A100339 A100340 A100341

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Nov 17 2004

STATUS

approved