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.

A248322 Egyptian fraction representation of sqrt(99) (A010550) using a greedy function.

Original entry on oeis.org

9, 2, 3, 9, 185, 40782, 1682066752, 6363269744807224762, 71990770113177468702243288679736023556, 7052581923050601721615256905785412578772858487621807510338728141989919040612
Offset: 0

Views

Author

Robert G. Wilson v, Oct 05 2014

Keywords

Crossrefs

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.
Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Programs

  • Mathematica
    Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 99]]

A010170 Continued fraction for sqrt(99).

Original entry on oeis.org

9, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			9.9498743710661995473447982... = 9 + 1/(1 + 1/(18 + 1/(1 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 12 2009

Links

Crossrefs

Cf. A010550 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[Sqrt[99],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011 *)
PadRight[{9},120,{18,1}] (* Harvey P. Dale, Aug 30 2021 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 27000); x=contfrac(sqrt(99)); for (n=0, 20000, write("b010170.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 12 2009

Formula

a(n+1) = 18^n mod 19, for all n >= 0. - M. F. Hasler, Mar 10 2011
From Amiram Eldar, Nov 14 2023: (Start)
Multiplicative with a(2^e) = 18, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 17/2^s). (End)

A041178 Numerators of continued fraction convergents to sqrt(99).

Original entry on oeis.org

9, 10, 189, 199, 3771, 3970, 75231, 79201, 1500849, 1580050, 29941749, 31521799, 597334131, 628855930, 11916740871, 12545596801, 237737483289, 250283080090, 4742832924909, 4993116004999, 94618921014891, 99612037019890, 1887635587372911, 1987247624392801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010550, A041179.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[99], 30]] (* Vincenzo Librandi, Oct 30 2013 *)
LinearRecurrence[{0,20,0,-1},{9,10,189,199},40] (* Harvey P. Dale, Jan 23 2019 *)

Formula

G.f.: -(x^3-9*x^2-10*x-9) / (x^4-20*x^2+1). - Colin Barker, Nov 05 2013

Extensions

More terms from Colin Barker, Nov 05 2013

A041179 Denominators of continued fraction convergents to sqrt(99).

Original entry on oeis.org

1, 1, 19, 20, 379, 399, 7561, 7960, 150841, 158801, 3009259, 3168060, 60034339, 63202399, 1197677521, 1260879920, 23893516081, 25154396001, 476672644099, 501827040100, 9509559365899, 10011386405999, 189714514673881, 199725901079880, 3784780734111721
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 18 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Links

Crossrefs

Cf. A041178, A010170, A020856, A010550, A002530.

Programs

  • Magma
    I:=[1, 1, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013
  • Mathematica
    Denominator[Convergents[Sqrt[99], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

Formula

G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - Colin Barker, Nov 14 2013
a(n) = 20*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 12 2013
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 18 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 18*a(2*n) + a(2*n - 1). (End)

Extensions

More terms from Colin Barker, Nov 14 2013
Showing 1-4 of 4 results.

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

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