A100864 -id:A100864 - cp's OEIS frontend

A100865 Records in the continued fraction expansion A100864.

1, 4, 74, 8457, 186282390, 430917181166219, 41151315877490090952542206046, 13991468824374967392702752173757116934238293984253807017

Offset: 1

Paul D. Hanna, Nov 21 2004

nonn

These terms are doubly exponential. The next term has 106 digits. Positions of these large partial quotients are given by A100866. Hans Havermann has observed that the ratio of number-of-digits to position number is about 1.03, nearly equal to the reciprocal of Lochs constant.

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.

Cf. A006519, A100338, A100863, A100864, A100866.

A100866 Positions of records in the continued fraction expansion A100864.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 35, 71, 143, 291, 635, 1407, 2979, 6101, 12339, 25019, 50413, 101339, 202793, 405745, 811365, 1624043, 3249293, 6502711, 13011309

Offset: 1

Paul D. Hanna, Nov 21 2004

nonn

Records form A100865 and are doubly exponential. These terms were independently computed by Robert G. Wilson v and Hans Havermann.

Cf. A100338, A100863, A100864, A100865.

PARI

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

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Nov 17 2004

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.

			1.353871128429882374388894084016608124227333416812118556923672649787001842...

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.

Cf. A006519, A100863, A100864, A100865.

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

/* 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}

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

A100863 Decimal expansion of the square of the constant (A100338) which has the continued fraction expansion equal to A006519 (highest power of 2 dividing n).

Original entry on oeis.org

1, 8, 3, 2, 9, 6, 7, 0, 3, 2, 3, 9, 6, 0, 0, 3, 0, 5, 4, 4, 2, 7, 2, 1, 9, 5, 4, 4, 2, 1, 0, 4, 1, 7, 3, 2, 4, 0, 5, 7, 7, 1, 6, 5, 6, 3, 2, 2, 7, 2, 1, 6, 8, 9, 7, 7, 9, 8, 3, 8, 9, 7, 7, 8, 5, 5, 7, 1, 8, 7, 9, 9, 0, 0, 7, 9, 0, 4, 7, 9, 4, 0, 3, 0, 8, 2, 8, 7, 8, 8, 7, 7, 0, 2, 8, 0, 8, 9, 4, 6, 7, 9, 6, 5, 4

Offset: 1

Paul D. Hanna, Nov 20 2004

The continued fraction of this constant (A100864) has large partial quotients (A100865) that appear to be doubly exponential.

			1.83296703239600305442721954421041732405771656322721689779838977855718799...

Cf. A006519, A100338, A100864, A100865.

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

cp's OEIS Frontend

A100865 Records in the continued fraction expansion A100864.

Views

Author

Keywords

Comments

Links

Crossrefs

A100866 Positions of records in the continued fraction expansion A100864.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A100863 Decimal expansion of the square of the constant (A100338) which has the continued fraction expansion equal to A006519 (highest power of 2 dividing n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI