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 1-5 of 5 results.

A040132 Continued fraction for sqrt(145).

Original entry on oeis.org

12, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			12 + 1/(24 + 1/(24 + 1/(24 + 1/(24 + ...)))) = sqrt(145).

Links

Crossrefs

Cf. A041264/A041265 (convergents), A176910 (decimal expansion).
Cf. A040000, A040002, A040006.

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[145],300] (* Vladimir Joseph Stephan Orlovsky, Mar 13 2011*)

Formula

From Elmo R. Oliveira, Feb 12 2024: (Start)
a(n) = 24 for n >= 1.
G.f.: 12*(1+x)/(1-x).
E.g.f.: 24*exp(x) - 12.
a(n) = 12*A040000(n) = 6*A040002(n) = 4*A040006(n). (End)

A041265 Denominators of continued fraction convergents to sqrt(145).

Original entry on oeis.org

1, 24, 577, 13872, 333505, 8017992, 192765313, 4634385504, 111418017409, 2678666803320, 64399421297089, 1548264777933456, 37222754091700033, 894894362978734248, 21514687465581321985, 517247393536930461888, 12435452132351912407297
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Michael A. Allen, May 04 2023: (Start)
Also called the 24-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 24 kinds of squares available. (End)

Links

Crossrefs

Cf. A041264, A176910, A040132.
Row n=24 of A073133, A172236 and A352361 and column k=24 of A157103.

Programs

  • Mathematica
    Denominator[Convergents[Sqrt[145], 30]] (* Vincenzo Librandi, Dec 14 2013 *)

Formula

a(n) = F(n, 24), the n-th Fibonacci polynomial evaluated at x=24. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 24*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=24.
G.f.: 1/(1-24*x-x^2). (End)

Extensions

More terms from Colin Barker, Nov 14 2013

A041269 Denominators of continued fraction convergents to sqrt(147).

Original entry on oeis.org

1, 8, 193, 1552, 37441, 301080, 7263361, 58407968, 1409054593, 11330844712, 273349327681, 2198125466160, 53028360515521, 426425009590328, 10287228590683393, 82724253735057472, 1995669318232062721, 16048078799591559240, 387149560508429484481
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041268, A176910, A040132.

Programs

  • Magma
    I:=[1,8,193,1552]; [n le 4 select I[n] else 194*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 14 2013
  • Mathematica
    Denominator[Convergents[Sqrt[147], 30]] (* Vincenzo Librandi, Dec 14 2013 *)

Formula

G.f.: -(x^2-8*x-1) / ((x^2-14*x+1)*(x^2+14*x+1)). - Colin Barker, Nov 14 2013
a(n) = 194*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 14 2013

Extensions

More terms from Colin Barker, Nov 14 2013

A176907 Decimal expansion of (9+sqrt(145))/16.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 28 2010

Keywords

Comments

Continued fraction expansion of (9+sqrt(145))/16 is A130793.

Examples

			(9+sqrt(145))/16 = 1.31509966117451846750...

Crossrefs

Cf. A176910 (decimal expansion of sqrt(145)), A130793 (repeat 1, 3, 5).

A176908 Decimal expansion of (7+sqrt(145))/16.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 28 2010

Keywords

Comments

Continued fraction expansion of (7+sqrt(145))/16 is A130794.

Examples

			(7+sqrt(145))/16 = 1.19009966117451846750...

Crossrefs

Cf. A176910 (decimal expansion of sqrt(145)), A130794 (repeat 1, 5, 3).

Programs

  • Mathematica
    RealDigits[(7+Sqrt[145])/16,10,120][[1]] (* Harvey P. Dale, Jan 06 2013 *)
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.