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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A099453 Expansion of 1/(1 - 7*x + 11*x^2).

Original entry on oeis.org

1, 7, 38, 189, 905, 4256, 19837, 92043, 426094, 1970185, 9104261, 42057792, 194257673, 897167999, 4143341590, 19134543141, 88365044497, 408075336928, 1884511869029, 8702754376995, 40189650079646, 185597252410577, 857094615997933, 3958092535469184, 18278606972307025
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

Associated to the knot 8_12 by the modified Chebyshev transform A(x)-> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099454 and A099455.

Crossrefs

Programs

  • GAP
    a:=[1,7];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019
  • Magma
    I:=[1,7]; [n le 2 select I[n] else 7*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019
    
  • Mathematica
    LinearRecurrence[{7,-11}, {1,7}, 30] (* G. C. Greubel, May 21 2019 *)
  • PARI
    Vec(1/(1-7*x+11*x^2) + O(x^30)) \\ Michel Marcus, Sep 09 2017
    
  • Sage
    [lucas_number1(n,7,11) for n in range(1, 30)] # Zerinvary Lajos, Apr 23 2009
    

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-11)^k*7^(n-2k).
a(n) = ((7+sqrt(5))^n - (7-sqrt(5))^n)/(2^n*sqrt(5)), n > 0. Binomial transform of A030191 (Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2); 3rd binomial transform of Fibonacci(n). - Creighton Dement, Apr 19 2005
a(n) = 7*a(n-1) - 11*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011
E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, May 13 2024

A099454 A Chebyshev transform of A099453 associated to the knot 8_12.

Original entry on oeis.org

1, 7, 37, 175, 792, 3521, 15539, 68369, 300431, 1319472, 5793745, 25437727, 111681277, 490315231, 2152620360, 9450575729, 41490490763, 182153978153, 799702876895, 3510901281888, 15413758929889, 67670362004791, 297090274041301, 1304302623454159, 5726223576745848
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 8_12. The g.f. is the image of the g.f. of A099453 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[(1+x^2)/(1-7x+13x^2-7x^3+x^4),{x,0,24}],x] (* Stefano Spezia, May 13 2024 *)

Formula

G.f.: (1+x^2)/(1-7x+13x^2-7x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*Sum{j=0..n-2*k} C(n-2*k-j, j)*(-11)^j*7^(n-2*k-2*j).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099453(n-2*k).
a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099453(k)/2.
a(n) = Sum_{k=0..n} A099455(n-k)*binomial(1, k/2)*(1+(-1)^k)/2.
Showing 1-2 of 2 results.