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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A159668 Expansion of (1 - x)/(1 - 28*x + x^2).

Original entry on oeis.org

1, 27, 755, 21113, 590409, 16510339, 461699083, 12911063985, 361048092497, 10096435525931, 282339146633571, 7895399670214057, 220788851619360025, 6174192445671866643, 172656599627192905979, 4828210597115729500769, 135017240119613233115553
Offset: 0

Views

Author

Paul Weisenhorn, Apr 19 2009

Keywords

Comments

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j) + 1 = a(j)*a(j) and 15*n(j) + 1 = b(j)*b(j) with positive integer numbers.
Positive values of x (or y) satisfying x^2 - 28*x*y + y^2 + 26 = 0. - Colin Barker, Feb 23 2014

Links

Crossrefs

Cf. A097311, A157456, A159669, A159673.
Cf. similar sequences listed in A238379.

Programs

  • Magma
    [n le 2 select 27^(n-1) else 28*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 25 2014
  • Maple
    for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq(simplify(ChebyshevU(n,14) -ChebyshevU(n-1,14)), n=0..40); # G. C. Greubel, Sep 26 2022
  • Mathematica
    CoefficientList[Series[(1-x)/(1-28x+x^2), {x,0,40}], x] (* Vincenzo Librandi, Feb 25 2014 *)
LinearRecurrence[{28,-1},{1,27},40] (* Harvey P. Dale, Apr 09 2014 *)
  • PARI
    Vec((-x+1)/(x^2-28*x+1) + O(x^100)) \\ Colin Barker, Feb 23 2014
  • SageMath
    def A159668(n): return chebyshev_U(n,14) - chebyshev_U(n-1,14)
[A159668(n) for n in range(40)] # G. C. Greubel, Sep 26 2022

Formula

G.f.: (1 - x)/(1 - 28*x + x^2).
The a(j) recurrence is a(0)=1, a(1)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (this sequence).
The b(j) recurrence is b(0)=1, b(1)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (A159669).
The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (A159673).
a(n) = (1/30)*(15-sqrt(195))*(1+(14+sqrt(195))^(2*n+1))/(14+sqrt(195))^n. - Bruno Berselli, Feb 25 2014
a(n) = 28*a(n-1) - a(n-2), a(0)=1, a(1)=27. - Harvey P. Dale, Apr 09 2014
a(n) = A097311(n) - A097311(n-1). - G. C. Greubel, Sep 26 2022

Extensions

More terms from Colin Barker, Feb 23 2014
New name and offset changed to 0 from Joerg Arndt, Feb 23 2014

A159673 Expansion of 56*x^2/(1 - 783*x + 783*x^2 - x^3).

Original entry on oeis.org

0, 56, 43848, 34289136, 26814060560, 20968561068840, 16397387941772376, 12822736401904929248, 10027363468901712899616, 7841385409944737582570520, 6131953363213315887857247080, 4795179688647403079566784646096, 3749824384568905994905337736000048
Offset: 1

Views

Author

Paul Weisenhorn, Apr 19 2009

Keywords

Comments

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j) + 1 = a(j)*a(j) and 15*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Links

Crossrefs

Cf. A157456, A159668, A159669.

Programs

  • Magma
    b:= func< n | Evaluate(ChebyshevSecond(n),391) >;
[(14/195)*(-1 +b(n+1) -781*b(n)): n in [1..30]]; // G. C. Greubel, Sep 25 2022
  • Maple
    for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq((14/195)*(simplify(ChebyshevU(n, 391) -781*ChebyshevU(n-1, 391)) -1), n=1..30); # G. C. Greubel, Sep 25 2022
  • Mathematica
    CoefficientList[Series[56 x/(- x^3 + 783 x^2 - 783 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{783,-783,1},{0,56,43848},20] (* Harvey P. Dale, Jan 06 2019 *)
  • PARI
    Vec(56*x^2/(-x^3+783*x^2-783*x+1) + O(x^100)) \\ Colin Barker, Feb 24 2014
  • PARI
    a(n) = round(-((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390) \\ Colin Barker, Jul 25 2016
  • SageMath
    def A159673(n): return (14/195)*(-1 + chebyshev_U(n, 391) - 781*chebyshev_U(n-1, 391))
[A159673(n) for n in range(1,30)] # G. C. Greubel, Sep 25 2022

Formula

The a(j) recurrence is a(1)=1, a(2)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (A159668).
The b(j) recurrence is b(1)=1, b(2)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (A159669).
The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (this sequence).
G.f.: 56*x^2/((1-x)*(1 - 782*x + x^2)). - Vincenzo Librandi, Feb 26 2014
a(n) = -((391+28*sqrt(195))^(-n)*(-1+(391+28*sqrt(195))^n)*(14+sqrt(195)+(-14+sqrt(195))*(391+28*sqrt(195))^n))/390. - Colin Barker, Jul 25 2016
a(n) = (14/195)*(-1 + ChebyshevU(n, 391) - 781*ChebyshevU(n-1, 391)). - G. C. Greubel, Sep 25 2022

Extensions

More terms and new name from Colin Barker, Feb 24 2014
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