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 1-3 of 3 results.

A010530 Decimal expansion of square root of 78.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Continued fraction expansion is 8 followed by {1, 4, 1, 16} repeated. - Harry J. Smith, Jun 09 2009

Examples

			8.831760866327846854764042726959253964174639480931417826210202972557139...

Links

Crossrefs

Cf. A010156 Continued fraction.

Programs

  • Mathematica
    RealDigits[N[78^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)
  • PARI
    { default(realprecision, 20080); x=sqrt(78); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010530.txt", n, " ", d)); } \\ Harry J. Smith, Jun 09 2009

A146160 Period 4: repeat [1, 4, 1, 16].

Original entry on oeis.org

1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1
Offset: 1

Views

Author

Artur Jasinski, Oct 27 2008

Keywords

Links

Crossrefs

Cf. A010156, A145996. [Artur Jasinski, Oct 29 2008]

Programs

  • Magma
    &cat[[1,4,1,16]^^20]; // Vincenzo Librandi, Feb 04 2016
  • Maple
    A146160:=n->[1, 4, 1, 16][(n mod 4)+1]: seq(A146160(n), n=0..100); # Wesley Ivan Hurt, Jun 15 2016
  • Mathematica
    Table[GCD[4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2], {k, 100}]
PadRight[{},100,{1,4,1,16}] (* or *) LinearRecurrence[{0,0,0,1},{1,4,1,16},90] (* Harvey P. Dale, Mar 29 2025 *)
  • PARI
    Vec((1+4*x+x^2+16*x^3)/(1-x^4) + O(x^100)) \\ Altug Alkan, Feb 04 2016

Formula

Continued fraction of (8 + sqrt(78))/14.
GCD(4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2) for k = 1,2,3,...
From Artur Jasinski, Oct 29 2008: (Start)
a(n) = 1 when n congruent to 1 or 3 mod 4.
a(n) = 4 when n congruent to 2 mod 4.
a(n) = 16 when n congruent to 0 mod 4. (End)
From Richard Choulet, Nov 03 2008: (Start)
a(n+4) = a(n).
a(n) = (9/2)*(-1)^n + (11/2) + 6*cos(Pi*n/2).
O.g.f.: f(z) = a(0)+a(1)*z+... = (1+4*z+z^2+16*z^3)/(1-z^4). (End)
E.g.f.: sinh(x) + 20*(sinh(x/2))^2 - 12*(sin(x/2))^2. - G. C. Greubel, Feb 03 2016
a(n) = a(-n). - Wesley Ivan Hurt, Jun 15 2016
a(n) = A109008(n)^2. - R. J. Mathar, Feb 12 2019
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2) = 4, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(12/4^s+3/2^s+1). (End)

Extensions

Choulet formula adapted for offset 1 from Wesley Ivan Hurt, Jun 15 2016

A041139 Denominators of continued fraction convergents to sqrt(78).

Original entry on oeis.org

1, 1, 5, 6, 101, 107, 529, 636, 10705, 11341, 56069, 67410, 1134629, 1202039, 5942785, 7144824, 120259969, 127404793, 629879141, 757283934, 12746422085, 13503706019, 66761246161, 80264952180, 1351000481041, 1431265433221, 7076062213925, 8507327647146
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041138, A010156, A020835, A010530.

Programs

  • Magma
    I:=[1,1,5,6,101,107,529,636]; [n le 8 select I[n] else 106*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
  • Mathematica
    Denominator/@Convergents[Sqrt[78], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^2 - x - 1) (x^4 + 6*x^2 + 1)/(x^8 - 106 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,106,0,0,0,-1},{1,1,5,6,101,107,529,636},30] (* Harvey P. Dale, Sep 15 2018 *)

Formula

G.f.: -(x^2-x-1)*(x^4+6*x^2+1) / (x^8-106*x^4+1). - Colin Barker, Nov 13 2013
a(n) = 106*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013
Showing 1-3 of 3 results.

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

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