cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A085395 Denominators of convergents to the Thue-Morse constant 0.41245403364...

Original entry on oeis.org

1, 2, 5, 12, 17, 80, 257, 1365, 2987, 4352, 20395, 45142, 65537, 372827, 16469925, 16842752, 83840933, 100683685, 285208303, 1241516897, 1526725200, 2768242097, 4294967297, 24243078582, 343698067445, 367941146027, 18740755368795
Offset: 1

Views

Author

Gary W. Adamson, Jun 27 2003

Keywords

Examples

			[2, 2, 2, 1, 4] = 33/80 = 0.4125.

Crossrefs

Companion numerators are A085394.
Cf. A085394, A014572, A014571.

Programs

  • Mathematica
    mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt]], {n, 0, 7}]; d = RealDigits[ N[ ToExpression[mt], 2^7]][[1]]; a = 0; Do[ a = a + N[ d[[n]]/2^(n + 1), 100], {n, 1, 2^7}]; f[n_] := FromContinuedFraction[ ContinuedFraction[a, n]]; Table[ Denominator[f[n]], {n, 1, 28}]

Formula

Write the convergents directly underneath the partial quotients (A014572) for 0.412454033... starting with the first partial quotient, 2: [2, 2, 2, 1, 4, 3, 5, 2, 1, 4, ...] such that [2] = 1/2, [2, 2] = 2/5, [ 2, 2, 2] = 5/12 and so on, the convergents being: 1/2, 2/5, 5/12, 7/17, 33/80, 106, 257, 563/1365, 1232/2987, 1795/4352, 8412/20395, ...

Extensions

Edited by Robert G. Wilson v, Jul 15 2003

A162634 Numerators of fractions with denominators A000215(n) approximating the Thue-Morse constant.

Original entry on oeis.org

1, 2, 7, 106, 27031, 1771476586, 7608434000728254871, 140350834813144189858090274002849666666, 47758914269546354982683078068829456704164423862093743397580034411621752859031
Offset: 0

Views

Author

Vladimir Shevelev, Jul 08 2009, Jul 14 2009

Keywords

Comments

One can prove that if in the sequence of numbers N for which A010060(N+2^n)= A010060(N) you replace the odious (evil) terms by 1's (0's), then we obtain 2^(n+1)-periodic (0,1)-sequence; if you write it in the form .xx...,i.e., as a binary infinite fraction, then the corresponding fraction has the form a(n)/A000215(n). These fractions very fast converge to the Thue-Morse constant .4124540336401...; e.g a(5)/(2^32+1) approximates this constant up to 10^(-9). These approximations differ from A074072-A074073. Conjecture. For n>=1, the fraction a(n)/A000215(n) is a convergent corresponding to the continued fraction for the Thue-Morse constant.

Links

Crossrefs

Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A074072, A074073.

Programs

  • PARI
    a(n)=if(n<=1, [1,2][n+1], 1+(2^(2^(n-1))-1)*a(n-1)); /* Joerg Arndt, Mar 11 2013 */

Formula

a(1)=2, and, for n>=2, a(n) = 1 + (2^(2^(n-1))-1) * a(n-1).

Extensions

Added more terms, Joerg Arndt, Mar 11 2013

A085396 Numerator and denominator sums of convergents to the Thue-Morse constant, 0.412454033...

Original entry on oeis.org

1, 3, 7, 17, 24, 113, 363, 1928, 4219, 6147, 28807, 63761, 92568, 526601, 23263012, 23789613, 118421464, 142211077, 402843618, 1753585549, 2156429167, 3910014716, 6066443883, 34242234131, 485457721717, 519699955848
Offset: 1

Views

Author

Gary W. Adamson, Jun 27 2003

Keywords

Comments

Let k = 0.412454..., then A085396(n)/A085394(n) [i.e., (numerator + denominator)/(numerator)] converges upon 3.424512... as n approaches infinity, where 3.424... = (k+1)/k. A085396(n)/A085395(n) [i.e., (numerator + denominator)/(denominator)], converges upon k+1, = 1.412454... Check: A085396(6)/A085394(6) = 363/106 = 3.4245...; while A085396(6)/A085395(6) = 393/257 = 1.41245... The constants (k+1) and (k+1)/k are generators for the Beatty pairs for the Thue-Morse constant, where the pairs are [(n*(k+1), (n*(k+1)/k], n = 1,2,3,...

Examples

			Convergents to the Thue-Morse constant 0.4124540336... are derived from continued fraction form shown in A014572, starting with A014572(1) = 2; then 0.412454... = [2, 2, 2, 1, 4, 3, 5, 2, 1, ...] (A014572). Example [2] = 1/2, [2,2] = 2/5, [2,2,2] = 5/12 and so on.

Crossrefs

Cf. A014571, A014572.

Programs

  • Mathematica
    mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt]], {n, 0, 7}];
d = RealDigits[ N[ ToExpression[mt], 2^7]][[1]];
a = 0; Do[ a = a + N[ d[[n]]/2^(n + 1), 100], {n, 1, 2^7}];
f[n_] := FromContinuedFraction[ ContinuedFraction[a, n]];
Table[ Numerator[ f[n]] + Denominator[ f[n]], {n, 2, 27}]

Formula

a(n) = A085394(n) + A085395(n) = numerator and denominator sums for convergents of 0.412454..., the convergents being 1/2, 2/5, 5/12, 7/17, 33/80, 106/257, 563/1365, 1232/2987, 1795/4352, 8412/20395, ...

Extensions

Edited by Robert G. Wilson v, Jul 15 2003

A162647 Numerators associated with denominators A000215(n) approximating the complementary Thue-Morse constant.

Original entry on oeis.org

2, 3, 10, 151, 38506, 2523490711, 10838310072981296746, 199931532107794273605284333428918544791, 68033174967769840440887906939858451149105560803546820641877549596291376780906
Offset: 0

Views

Author

Vladimir Shevelev, Jul 08 2009, Jul 14 2009

Keywords

Comments

If in the sequence of numbers N for which A010060(N+2^n)=1-A010060(N) the odious (evil) terms are
replaced by 1's (0's), we obtain a 2^(n+1)-periodic binary sequence. These are the post-period
binary (base-2) digits of the complementary Thue-Morse constant 1-A014571 = 0.58754596635989240221663...,
which has a continued fraction and convergents 3/5, 7/12, 10/17, 47/80, 151/257, 801/1365,...
The a(n) are numerators of the convergents selected with denominators taken from A000215.

Links

Crossrefs

Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A162634.

Formula

a(n)=A000215(n)-A162634(n). For n>=1, a(n+1)=1+(2^(2^n)-1)*a(n) = 1+A051179(n)*a(n).

Extensions

Edited by R. J. Mathar, Sep 23 2009
Showing 1-4 of 4 results.

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