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

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• sign

Comments

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.

Links

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

Programs

Mathematica

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

Sage

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

Formula

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