A099451 -id:A099451 - 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

A099452 An Alexander sequence for the knot 7_7.

Original entry on oeis.org

1, 5, 16, 40, 79, 110, 23, -520, -2336, -6995, -16574, -31075, -38848, 9560, 258631, 1043950, 2978719, 6781640, 12060848, 13119125, -12022526, -124662155, -461573264, -1259138680, -2752822273, -4615067410, -4134056729, 8360350360, 58685747584, 202130368445, 528415922498

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 7_7. 1/(1-5*x+9*x^2-5*x^3+x^4) is the image of the g.f. of A099450 under the modified Chebyshev transform A(x)->(1/(1+x^2)^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).

LinearRecurrence[{5,-9,5,-1},{1,5,16,40,79},40] (* Harvey P. Dale, Apr 18 2019 *)

G.f.: (1-x)*(1+x)*(1+x^2)/(1-5*x+9*x^2-5*x^3+x^4). - corrected by R. J. Mathar, Nov 24 2012

a(n)=A099451(n)-A099451(n-2).

cp's OEIS Frontend

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

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