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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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)

A099461 An Alexander sequence for the knot 9_48.

Original entry on oeis.org

1, 7, 38, 196, 1001, 5110, 26093, 133252, 680510, 3475339, 17748434, 90640627, 462898478, 2364006148, 12072895733, 61655851222, 314874250049, 1608051650884, 8212262868470, 41939735818687, 214184746483778, 1093833919809295, 5586171115205846, 28528378178106436, 145693417671662033, 744051127629095062, 3799842775146922277, 19405662567631938052, 99104031922539424718
Offset: 0

Views

Author

Paul Barry, Oct 16 2004

Keywords

Comments

The denominator 1 -7*x +11*x^2 -7*x^3 +x^4 is a parameterization of the Alexander polynomial for the knot 9_48. 1/(1 -7*x +11*x^2 -7*x^3 +x^4) is the image of the g.f. of A099459 under the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)).

Links

Crossrefs

Cf. A099459, A099460.

Programs

  • Magma
    I:=[7,38,196,1001]; [1] cat [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..41]]; // G. C. Greubel, Nov 18 2021
  • Mathematica
    LinearRecurrence[{7,-11,7,-1},{1,7,38,196,1001},40] (* Harvey P. Dale, Jun 18 2021 *)
  • Sage
    def A099461_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
A099461_list(40) # G. C. Greubel, Nov 18 2021

Formula

a(n) = A099460(n) - A099460(n-2).
G.f.: (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4). - Corrected by R. J. Mathar, Nov 23 2012
Showing 1-2 of 2 results.

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