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.

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A378345 Continued fraction expansion of the base 4 Champernowne constant.

Original entry on oeis.org

0, 2, 2, 1, 7, 1, 1, 2, 1, 1, 1, 1, 6806293849, 1, 33, 157, 1, 2, 1, 3, 1, 1, 2345427263108642344323518197756649380964709224412095403124301722165, 2, 2, 1, 1, 1, 3, 1, 1, 6, 2, 7, 11, 1, 1, 7, 12, 1, 1, 1, 126, 3, 13, 1, 13, 4, 33, 3, 1, 1, 1, 3, 2, 4, 1, 9, 2
Offset: 0

Views

Author

Joshua Searle, Dec 13 2024

Keywords

Links

Crossrefs

Cf. A030373 (base 4 expansion), A378328 (decimal expansion).
Other continued fractions: A066717, A077772, A378346, A378347, A378348, A378349, A378350, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[4], 100]

A378346 Continued fraction expansion of the base 5 Champernowne constant.

Original entry on oeis.org

0, 3, 4, 1, 1, 2, 2, 18, 1, 20, 1302701925685142513155, 3, 5, 6, 1, 1, 1, 1, 1, 1, 2, 13, 5, 2, 1, 22, 1, 1
Offset: 0

Views

Author

Joshua Searle, Dec 13 2024

Keywords

Comments

The next term a(28) is approximately equal to 2.83 * 10^173.

Links

Crossrefs

Cf. A031219 (base 5 expansion), A378329 (decimal exapansion).
Other continued fractions: A066717, A077772, A378345, A378347, A378348, A378349, A378350, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[5], 100]

A378347 Continued fraction expansion of the base 6 Champernowne constant.

Original entry on oeis.org

0, 4, 5, 1, 10, 1, 4, 3, 9, 1, 2, 2, 1, 1, 699745284439054751106354294914368414245, 2, 5, 1, 20, 22, 2, 2, 1, 10, 3, 1, 2, 2, 2, 1, 1, 2, 1, 1
Offset: 0

Views

Author

Joshua Searle, Dec 13 2024

Keywords

Comments

The next term a(34) is approximately equal to 1.21 * 10^364.

Links

Crossrefs

Cf. A030548 (base 6 expansion), A378330 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378348, A378349, A378350, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[6], 100]

A378348 Continued fraction expansion of the base 7 Champernowne constant.

Original entry on oeis.org

0, 5, 6, 1, 85, 1, 2, 1, 11, 1, 3, 2, 1, 5, 1, 2, 8697444597678755989498288581049684565698396369776180853037564, 1, 4, 2, 8, 6, 1, 2, 11, 1, 11, 1, 9, 2, 11, 1, 13, 2, 3, 10
Offset: 0

Views

Author

Joshua Searle, Dec 14 2024

Keywords

Comments

The next term a(36) is approximately equal to 4.24*10^662.

Links

Crossrefs

Cf. A030998 (base 7 expansion), A378331 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378349, A378350, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[7], 100]

A378349 Continued fraction expansion of the base 8 Champernowne constant.

Original entry on oeis.org

0, 6, 7, 1, 842, 5, 11, 2, 1, 4, 1, 12, 1217611913245203113561611289624720261608646275831638269345353220034950193075766082779756144, 39, 1, 13, 19, 1, 1, 2, 1, 6, 1, 4, 9, 1, 2, 1, 3, 2, 1, 223, 2, 1
Offset: 0

Views

Author

Joshua Searle, Dec 14 2024

Keywords

Comments

The next term a(34) is approximately equal to 5.28 * 10^1099.

Links

Crossrefs

Cf. A054634 (base 8 expansion), A378332 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378348, A378350, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[8], 100]

A378350 Continued fraction expansion of the base 9 Champernowne constant.

Original entry on oeis.org

0, 7, 8, 1, 10222, 1, 1, 1, 1, 1, 12, 1, 1, 1, 145, 1, 13127841267973253934598674824559230051317913195904874825561053745645554655306632773083671838234108227370808367172269493508107, 1, 7, 3, 1, 1, 1, 2, 2, 15, 3, 2, 1, 3, 2, 1, 1, 7, 4, 1, 4, 1, 1, 3, 3, 1, 1
Offset: 0

Views

Author

Joshua Searle, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A031076 (base 9 expansion), A378333 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378348, A378349, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[9], 100]

A041007 Denominators of continued fraction convergents to sqrt(6).

Original entry on oeis.org

1, 2, 9, 20, 89, 198, 881, 1960, 8721, 19402, 86329, 192060, 854569, 1901198, 8459361, 18819920, 83739041, 186298002, 828931049, 1844160100, 8205571449, 18255302998, 81226783441, 180708869880
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

sqrt(6) = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721), ...; where sqrt(6) = 2.4494897427... and the sum of the first 5 terms of this series = 2.449489737... - Gary W. Adamson, Dec 21 2007
sqrt(6) = 2 + continued fraction [2, 4, 2, 4, 2, 4, ...] = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721) + ... - Gary W. Adamson, Dec 21 2007
Interspersion of 2 sequences, A072256 and 2*A004189. - Gerry Martens, Jun 10 2015
For n > 0, a(n) equals the permanent of the n X n tridiagonal matrix with the main diagonal alternating sequence [2, 4, 2, 4, ...] and 1's along the superdiagonal and the subdiagonal. - Rogério Serôdio, Apr 01 2018

Links

Crossrefs

Cf. A010464, A041006.
Cf. A072256, A004189.

Programs

Formula

G.f.: (1+2*x-x^2)/(1-10*x^2+x^4). - Colin Barker, Dec 31 2011
From Rogério Serôdio, Apr 01 2018: (Start)
Recurrence formula: a(n) = (3 + (-1)^n)*a(n-1) + a(n-2), a(0) = 1, a(1) = 2.
Some properties:
(1) a(n)^2 - a(n-2)^2 = (3+(-1)^n)*a(2*n-1), for n > 1;
(2) a(2*n+1) = a(n)*(a(n+1) + a(n-1)), for n > 0;
(3) a(2*n) = A142239(2*n), for n >= 0;
(4) a(2*n+1) = A041007(2*n+1)/2, for n >= 0;
(5) a(2*n-1)*A142239(2*n+1) = a(n)^2 - 1, for n > 0;
(6) a(2*n) = a(n)*A142239(n) + a(n-1)*A142239(n-1), for n > 0;
(7) Sum_{k=0..n} a(2*k+1)*(A142239(2*k) + A142239(2*(k+1))) = Sum_{k=0..n} a(3+4*k);
(8) Sum_{k=0..n} (a(2*k-1) + a(2*k+1))*A142239(2*k) = Sum_{k=0..n} A142239(3+4*k). (End)
a(n) = ((2 + sqrt(6))^(n+1) - (2 - sqrt(6))^(n+1))/(sqrt(6) * 2^(ceiling(n/2) + 1)). - Robert FERREOL, Oct 14 2024

A041226 Numerators of continued fraction convergents to sqrt(125).

Original entry on oeis.org

11, 56, 67, 123, 682, 15127, 76317, 91444, 167761, 930249, 20633239, 104096444, 124729683, 228826127, 1268860318, 28143753123, 141987625933, 170131379056, 312119004989, 1730726404001, 38388099893011, 193671225869056, 232059325762067, 425730551631123
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A001946.
For the terms of the periodical sequence of the continued fraction for sqrt(125) see A010186. We observe that its period is five. (End)

Links

Crossrefs

Cf. A041227, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[125], 30]] (* Vincenzo Librandi, Oct 31 2013 *)

Formula

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5n) = A001946(3n+1),
a(5n+1) = (A001946(3n+2) - A001946(3n+1))/2,
a(5n+2) = (A001946(3n+2) + A001946(3n+1))/2,
a(5n+3) = A001946(3n+2),
a(5n+4) = A001946(3n+3)/2. (End)
G.f.: -(x^9 -11*x^8 +56*x^7 -67*x^6 +123*x^5 +682*x^4 +123*x^3 +67*x^2 +56*x +11) / ((x^2 +4*x -1)*(x^4 -7*x^3 +19*x^2 -3*x +1)*(x^4 +3*x^3 +19*x^2 +7*x +1)). - Colin Barker, Nov 08 2013

Extensions

More terms from Colin Barker, Nov 08 2013

A041318 Numerators of continued fraction convergents to sqrt(173).

Original entry on oeis.org

13, 79, 92, 171, 1118, 29239, 176552, 205791, 382343, 2499849, 65378417, 394770351, 460148768, 854919119, 5589663482, 146186169651, 882706681388, 1028892851039, 1911599532427, 12498490045601, 326872340718053, 1973732534353919, 2300604875071972
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041319, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.
Cf. A010217 (continued fraction).

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[173], 30]] (* Vincenzo Librandi, Nov 01 2013 *)
LinearRecurrence[{0,0,0,0,2236,0,0,0,0,1},{13,79,92,171,1118,29239,176552,205791,382343,2499849},30] (* Harvey P. Dale, Jul 28 2018 *)

Formula

a(5*n) = A088316(3*n+1), a(5*n+1) = (A088316(3*n+2) - A088316(3*n+1))/2, a(5*n+2) = (A088316(3*n+2)+A088316(3*n+1))/2, a(5*n+3) = A088316(3*n+2) and a(5*n+4) = A088316(3*n+3)/2. [Johannes W. Meijer, Jun 12 2010]
G.f.: -(x^9-13*x^8+79*x^7-92*x^6+171*x^5+1118*x^4+171*x^3+92*x^2+79*x+13) / (x^10+2236*x^5-1). - Colin Barker, Nov 08 2013

Extensions

More terms from Colin Barker, Nov 08 2013

A041426 Numerators of continued fraction convergents to sqrt(229).

Original entry on oeis.org

15, 106, 121, 227, 1710, 51527, 362399, 413926, 776325, 5848201, 176222355, 1239404686, 1415627041, 2655031727, 20000849130, 602680505627, 4238764388519, 4841444894146, 9080209282665, 68402909872801, 2061167505466695, 14496575448139666, 16557742953606361
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A090301.
For the terms of the periodical sequence of the continued fraction for sqrt(229) see A040213. We observe that its period is five. (End)

Links

Crossrefs

Cf. A041427, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[229], 30]] (* Vincenzo Librandi, Nov 01 2013 *)
LinearRecurrence[{0,0,0,0,3420,0,0,0,0,1},{15,106,121,227,1710,51527,362399,413926,776325,5848201},30] (* Harvey P. Dale, Dec 19 2016 *)

Formula

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5n) = A090301(3n+1), a(5n+1) = (A090301(3n+2) - A090301(3n+1))/2, a(5n+2) = (A090301(3n+2) + A090301(3n+1))/2, a(5n+3) = A090301(3n+2) and a(5n+4) = A090301(3n+3)/2. (End)
G.f.: -(x^9-15*x^8+106*x^7-121*x^6+227*x^5+1710*x^4+227*x^3+121*x^2+106*x+15) / (x^10+3420*x^5-1). - Colin Barker, Nov 08 2013

Extensions

More terms from Colin Barker, Nov 08 2013
Previous Showing 51-60 of 2828 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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