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 1-2 of 2 results.

A099459 Expansion of 1/(1 - 7*x + 9*x^2).

Original entry on oeis.org

1, 7, 40, 217, 1159, 6160, 32689, 173383, 919480, 4875913, 25856071, 137109280, 727060321, 3855438727, 20444528200, 108412748857, 574888488199, 3048504677680, 16165536349969, 85722212350663, 454565659304920
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

Associated to the knot 9_48 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099460 and A099461.

Links

Crossrefs

Cf. A002450, A004187, A016153, A099460, A099461, A102902.

Programs

  • Magma
    [n le 2 select 7^(n-1) else 7*Self(n-1) -9*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 18 2021
  • Mathematica
    LinearRecurrence[{7,-9},{1,7},30] (* Harvey P. Dale, Jan 06 2012 *)
  • Sage
    [lucas_number1(n,7,9) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-9)^k*7^(n-2*k).
a(n) = Sum{k=0..n} binomial(2*n-k+1, k) * 3^k. - Paul Barry, Jan 17 2005
a(n) = 7*a(n-1) - 9*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011
a(n) = ((7 + sqrt(13))^(n+1) - (7 - sqrt(13))^(n+1))/(2^(n+1)*sqrt(13)). - Rolf Pleisch, May 19 2011
a(n) = 3^(n-1)*ChebyshevU(n-1, 7/6). - G. C. Greubel, Nov 18 2021
From Peter Bala, Jul 23 2025: (Start)
The following products telescope:
Product_{k >= 1} 1 + 3^k/a(k) = (1 + sqrt(13))/2.
Product_{k >= 1} 1 - 3^k/a(k) = (1 + sqrt(13))/14,
Product_{k >= 1} 1 + (-3)^k/a(k) = (13 + sqrt(13))/26.
Product_{k >= 1} 1 - (-3)^k/a(k) = (13 + sqrt(13))/14. (End)

A099460 A Chebyshev transform of A099459 associated to the knot 9_48.

Original entry on oeis.org

1, 7, 39, 203, 1040, 5313, 27133, 138565, 707643, 3613904, 18456077, 94254531, 481354555, 2458260679, 12554250288, 64114111901, 327428500337, 1672165762785, 8539691368807, 43611901581472, 222724437852585
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

The denominator is a parameterization of the Alexander polynomial for the knot 9_48. The g.f. is the image of the g.f. of A099459 under the Chebyshev transform A(x) -> (1/(1+x^2))*A(x/(1+x^2)).

Links

Crossrefs

Cf. A099459, A099461.

Programs

  • Magma
    I:=[1,7,39,203]; [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, Nov 18 2021
  • Mathematica
    LinearRecurrence[{7,-11,7,-1}, {1,7,39,203}, 30] (* G. C. Greubel, Nov 18 2021 *)
  • Sage
    def A099460_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
A099460_list(30) # G. C. Greubel, Nov 18 2021

Formula

G.f.: (1+x^2)/(1 -7*x +11*x^2 -7*x^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)(-9)^j*7^(n-2*k-2*j) ).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-1)^k*A099459(n-2*k).
a(n) = (1/2)*Sum_{k=0..n} (-1)^((n-k)/2)*(1 + (-1)^(n+k))*binomial((n+k)/2, k) *A099459(k).
a(n) = Sum_{k=0..n} A099461(n-k)*binomial(1, k/2)*((1+(-1)^k)/2).
Showing 1-2 of 2 results.

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