A099452 -id:A099452 - cp's OEIS frontend

A099450 Expansion of 1/(1 - 5x + 7x^2).

1, 5, 18, 55, 149, 360, 757, 1265, 1026, -3725, -25807, -102960, -334151, -950035, -2411118, -5405345, -10148899, -12907080, 6506893, 122884025, 568871874, 1984171195, 5938752857, 15804565920, 37451559601, 76625836565, 120968265618, 68460472135, -504475498651

Offset: 0

Paul Barry, Oct 16 2004

Associated to the knot 7_7 by the modified Chebyshev transform A(x)-> (1/(1+x^2)^2)A(x/(1+x^2)). See A099451 and A099452.

Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (5,-7).

CoefficientList[Series[1/(1-5x+7x^2),{x,0,40}],x] (* or *) LinearRecurrence[ {5,-7},{1,5},40] (* Harvey P. Dale, Oct 21 2016 *)

[lucas_number1(n,5,7) for n in range(1, 30)] # Zerinvary Lajos, Apr 22 2009

a(n) = sum{k=0..floor(n/2), binomial(n-k, k)(-7)^k*5^(n-2k)}.

a(n) = 5*a(n-1) - 7*a(n-2), a(0)=1, a(1)=5. - Philippe Deléham, Nov 15 2008

A099451 A Chebyshev transform of A099450 associated to the knot 7_7.

Original entry on oeis.org

1, 5, 17, 45, 96, 155, 119, -365, -2217, -7360, -18791, -38435, -57639, -28875, 200992, 1015075, 3179711, 7796715, 15240559, 20915840, 3218033, -103746315, -458355231, -1362884995, -3211177504, -5977952405, -7345234233, 2382397955, 51340513351, 204512766400, 579756435849

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 7_7. The g.f. is the image of the g.f. of A099450 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,-1).

Cf. A099450, A099452.

G.f.: (1+x^2)/(1-5x+9x^2-5x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k * Sum_{j=0..n-2k} C(n-2k-j, j)*(-7)^j*5^(n-2k-2j).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099450(n-2k).
a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099450(k)/2.
a(n) = Sum_{k even, 0<=k<=n} A099452(n-k). [corrected by Kevin Ryde, Jul 24 2022]

cp's OEIS Frontend

A099450 Expansion of 1/(1 - 5x + 7x^2).

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