John K. Sikora - cp's OEIS frontend

A244759 Number of ternary digits in the n-th term of the continued fraction of the base 3 Champernowne constant (A077772).

0, 1, 1, 1, 4, 1, 5, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 33, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 2, 2, 6, 1, 139, 1, 1, 1, 2, 2, 3, 2, 1, 3, 1, 1, 2, 4, 1, 2, 2, 1, 4, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 1, 1, 1

Offset: 1

John K. Sikora, Jul 06 2014

John K. Sikora, Table of n, a(n) for n = 1..4956
J. K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
J. K. Sikora, Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)

Cf. A244332 (Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant.)

A244758 Number of binary digits in the n-th term of the continued fraction of the base-2 Champernowne constant (A066717).

Original entry on oeis.org

0, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3, 3, 1, 2, 9, 1, 3, 1, 1, 2, 9, 3, 2, 3, 1, 2, 1, 2, 1, 1, 23, 1, 3, 3, 1, 2, 1, 3, 1, 5, 1, 3, 1, 3, 3, 1, 3, 1, 3, 1, 2, 1, 3, 2, 2, 6, 2, 4, 1, 4, 2, 3, 2, 5, 53, 1, 1, 7, 1, 3, 3, 3, 3

Offset: 1

John K. Sikora, Jul 05 2014

John K. Sikora, Table of n, a(n) for n = 1..8174
John K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
John K. Sikora, Number of binary digits of the first 98093504 terms of the continued fraction of the base 2 Champernowne Constant (240 MB zipped)

Cf. A066717 (The continued fraction for the "binary" Champernowne constant.)

Cf. A244330 (Position of the incrementally largest term in the continued fraction for the base 2 Champernowne constant.)

A244757 Incrementally largest terms in the continued fraction for the base 3 Champernowne constant (A077772).

Original entry on oeis.org

0, 1, 2, 37, 162, 3068518062211324, 1079268324684171943515797470873767312825026176345571319042096689270

Offset: 1

John K. Sikora, Jul 05 2014

John K. Sikora, Table of n, a(n) for n = 1..7

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)

Cf. A244332 (Position of the incrementally largest term in the continued fraction for the ternary Champernowne constant.)

A244333 Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant (A077772).

Original entry on oeis.org

0, 1, 1, 4, 5, 33, 139, 515, 1809, 6181, 20759, 68871, 226333, 738089, 2391459, 7705867, 24711977, 78918957, 251105839

Offset: 1

John K. Sikora, Jun 27 2014

John K. Sikora, Table of n, a(n) for n = 1..19
J. K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
J. K. Sikora, Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)
Cf. A244757 (Incrementally largest terms in the continued fraction for the base 3 Champernowne constant.)
Cf. A244332 (Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant.)

puts (5..19).collect {|n| (1..(n-3)).inject(0) {|sum, m| sum+2*m*3**(m-1)}+3-n-2*((1..(n-4)).inject(0) {|sum1, m1| sum1+2*m1*3**(m1-1)}+3-(n-1))-3*(n-2)+4}

It appears that: Define NCD(N)=3-N+(sum{m=1..(N-3)} 2*m*3^(m-1)); then for n>=5, a(n) = NCD(n)-2*NCD(n-1)-3*(n-2)+4.

A244332 Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant (A077772).

Original entry on oeis.org

1, 2, 4, 5, 7, 19, 39, 121, 321, 813, 2063, 4957, 11731, 26329, 59501, 132273, 294475, 644557, 1392933, 2982557

Offset: 1

John K. Sikora, Jun 26 2014

John K. Sikora, Table of n, a(n) for n = 1..20
J. K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
J. K. Sikora, Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)
Cf. A244757 (Incrementally largest terms in the continued fraction for the base 3 Champernowne constant.)
Cf. A244333 (Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant.)

A244331 Number of binary digits in the high-water marks of the terms of the continued fraction of the base-2 Champernowne constant.

Original entry on oeis.org

0, 1, 3, 9, 23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, 65505, 131039, 262109, 524251, 1048537, 2097111, 4194261, 8388563, 16777169

Offset: 1

John K. Sikora, Jun 27 2014

Conjecture: partial sums of A296965 (equivalent to observation about A183155 below). - Sean A. Irvine, Jul 16 2022

John K. Sikora, Table of n, a(n) for n = 1..24
John K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
John K. Sikora, Number of binary digits of the first 98093504 terms of the continued fraction of the base 2 Champernowne Constant (240 MB zipped)

Cf. A066717, A066718, A244330.

Ruby
```
puts (4..24).collect{|n| 2**n-2*n+1}
```

puts (4..24).collect {|n| (1..n).inject(0) {|sum, m| sum+m*2**(m-1)}-n-2*((1..(n-1)).inject(0) {|sum1, m1| sum1+m1*2**(m1-1)}-(n-1))-3*n+4}

It appears that for n >= 4, a(n) = 2^n - 2*n + 1 = A183155(n-1).

Also it appears that if we define NCD(N) = (Sum_{m=1..N} m*2^(m-1)) - N, then for n >= 4, a(n) = NCD(n) - 2*NCD(n-1) - 3*n + 4.

A244330 Position of the incrementally largest term in the continued fraction for the base-2 Champernowne constant (A066717).

Original entry on oeis.org

1, 2, 3, 15, 31, 65, 153, 343, 775, 1729, 3735, 8175, 17371, 36447, 76077, 157471, 324801, 669469, 1370903, 2807215, 5728545, 11679949, 23777059, 48348375, 98093505

Offset: 1

John K. Sikora, Jun 25 2014

John K. Sikora, Table of n, a(n) for n = 1..25
John K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
John K. Sikora, Number of binary digits of the first 98093504 terms of the continued fraction of the base 2 Champernowne Constant (240 MB zipped)

Cf. A066717 (The continued fraction for the "binary" Champernowne constant.)
Cf. A066718 (Incrementally largest terms in the continued fraction for the "binary" Champernowne constant.)
Cf. A244331 (Number of binary digits in the high-water marks of the terms of the continued fraction of the base 2 Champernowne constant.)

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User: John K. Sikora

A244759 Number of ternary digits in the n-th term of the continued fraction of the base 3 Champernowne constant (A077772).

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A244758 Number of binary digits in the n-th term of the continued fraction of the base-2 Champernowne constant (A066717).

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A244757 Incrementally largest terms in the continued fraction for the base 3 Champernowne constant (A077772).

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A244333 Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant (A077772).

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Ruby

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A244332 Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant (A077772).

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Author

Keywords

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Crossrefs

A244331 Number of binary digits in the high-water marks of the terms of the continued fraction of the base-2 Champernowne constant.

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Author

Keywords

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Links

Crossrefs

Programs

Ruby

Ruby

Formula

A244330 Position of the incrementally largest term in the continued fraction for the base-2 Champernowne constant (A066717).

Views

Author

Keywords

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Crossrefs