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.

Previous Showing 21-24 of 24 results.

A382105 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372269.

Original entry on oeis.org

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

Views

Author

A.H.M. Smeets, Mar 27 2025

Keywords

Comments

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
k | zeros | corresponding weights
---+---------------------------+--------------------------
2 | A020760 | A000007*10
3 | A010513/10 | A010716
4 | A372267, A372268 | A382103, A382104
5 | A372269, A372270 | this sequence, A382106
6 | A372271, A372272, A372273 | A382107, A382686, A382687
7 | A372274, A372275, A372276 | A382688, A382689, A382690

Examples

			0.47862867049936646804129151483563819291229555334...

Links

Crossrefs

Cf. A372269.

Formula

Equals (322+13*sqrt(70))/900.

A040052 Continued fraction for sqrt(60).

Original entry on oeis.org

7, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			7.74596669241483377035853079... = 7 + 1/(1 + 1/(2 + 1/(1 + 1/(14 + ...)))). - _Harry J. Smith_, Jun 07 2009

Links

Crossrefs

Cf. A000007, A010513 (decimal expansion), A248285 (Egyptian fractions).

Programs

  • Magma
    [7] cat &cat[ [1, 2, 1, 14]: n in [1..18]]; // Bruno Berselli, Mar 07 2011
  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[60],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)
PadRight[{7},120,{14,1,2,1}] (* Harvey P. Dale, Aug 07 2019 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 19000); x=contfrac(sqrt(60)); for (n=0, 20000, write("b040052.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 07 2009

Formula

From Bruno Berselli, Mar 07 2011: (Start)
G.f.: (7 + x + 2*x^2 + x^3 + 7*x^4)/(1-x^4).
a(n) = (6*(-i)^n + 6*i^n + 7*(-1)^n + 9)/2 - 7*A000007(n), where i is the imaginary unit. (End)
From Amiram Eldar, Nov 13 2023: (Start)
Multiplicative with a(2) = 2, a(2^e) = 14 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 1/2^s + 3/4^(s-1)). (End)

A041105 Denominators of continued fraction convergents to sqrt(60).

Original entry on oeis.org

1, 1, 3, 4, 59, 63, 185, 248, 3657, 3905, 11467, 15372, 226675, 242047, 710769, 952816, 14050193, 15003009, 44056211, 59059220, 870885291, 929944511, 2730774313, 3660718824, 53980837849, 57641556673, 169263951195, 226905507868, 3345941061347, 3572846569215
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Interspersion of 4 linear recurrences with constant coefficients. - Gerry Martens, Jun 10 2015

Links

Crossrefs

Cf. A041104, A040052, A020817, A010513.
Cf. A258684.

Programs

  • Magma
    I:=[1, 1, 3, 4, 59, 63, 185, 248]; [n le 8 select I[n] else 62*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
  • Maple
    numtheory:-cfrac(sqrt(60),100,'con','den'):
den[1..-2]; # Robert Israel, Jun 09 2015
  • Mathematica
    Denominator[Convergents[Sqrt[60], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
d0 := LinearRecurrence[{62, -1}, {1, 59}, 20]
d1 := LinearRecurrence[{62, -1}, {1, 63}, 20] (* A258684  *)
d2 := LinearRecurrence[{62, -1}, {3, 185}, 20]
d3 := LinearRecurrence[{62, -1}, {4, 248}, 20]
Flatten[MapIndexed[{d0[[#]] , d1[[#]], d2[[#]] , d3[[#]]} &,
  Range[10]]] (* Gerry Martens, Jun 09 2015 *)
LinearRecurrence[{0, 0, 0, 62, 0, 0, 0, -1},{1, 1, 3, 4, 59, 63, 185, 248},30] (* Ray Chandler, Aug 03 2015 *)

Formula

G.f.: -(x^2-x-1)*(x^4+4*x^2+1) / ((x^4-8*x^2+1)*(x^4+8*x^2+1)). - Colin Barker, Nov 12 2013
a(n) = 62*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

Extensions

More terms from Colin Barker, Nov 12 2013

A041104 Numerators of continued fraction convergents to sqrt(60).

Original entry on oeis.org

7, 8, 23, 31, 457, 488, 1433, 1921, 28327, 30248, 88823, 119071, 1755817, 1874888, 5505593, 7380481, 108832327, 116212808, 341257943, 457470751, 6745848457, 7203319208, 21152486873, 28355806081, 418133772007, 446489578088, 1311112928183, 1757602506271
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Interspersion of 4 linear recurrences with constant coefficients. - Gerry Martens, Jun 10 2015

Links

Crossrefs

Cf. A010513, A041105.

Programs

  • Maple
    numtheory:-cfrac(sqrt(60),50,'con'):
map(numer,con[1..-2]); # Robert Israel, Jun 09 2015
  • Mathematica
    Numerator/@Convergents[Sqrt[60],30]  (* Harvey P. Dale, Apr 26 2011 *)
n0 := LinearRecurrence[{62, -1}, {7, 457}, 10]
n1 := LinearRecurrence[{62, -1}, {8, 488}, 10]
n2 := LinearRecurrence[{62, -1}, {23, 1433}, 10]
n3 := LinearRecurrence[{62, -1}, {31, 1921}, 10]
Flatten[MapIndexed[{n0[[#]],n1[[#]],n2[[#]],n3[[#]]} &, Range[10]]] (* Gerry Martens, Jun 09 2015 *)
LinearRecurrence[{0, 0, 0, 62, 0, 0, 0, -1},{7, 8, 23, 31, 457, 488, 1433, 1921},28] (* Ray Chandler, Aug 03 2015 *)

Formula

G.f.: -(x^7-7*x^6+8*x^5-23*x^4-31*x^3-23*x^2-8*x-7) / ((x^4-8*x^2+1)*(x^4+8*x^2+1)). - Colin Barker, Nov 05 2013

Extensions

More terms from Colin Barker, Nov 05 2013
Previous Showing 21-24 of 24 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.