A099450 Expansion of 1/(1 - 5*x + 7*x^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

0,2

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

Table of n, a(n) for n=0..28.

Dror Bar-Natan, The Rolfsen Knot Table.

Index entries for linear recurrences with constant coefficients, signature (5,-7).

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

E.g.f.: exp(5*x/2) * (cos(sqrt(3)*x/2) + 5*sin(sqrt(3)*x/2)/sqrt(3)). - Amiram Eldar, Feb 10 2026

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 *)

Prog

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

Crossrefs

Cf. A099451, A099452.

Sequence in context: A081492 A263318 A011845 * A360191 A344847 A145129

Adjacent sequences: A099447 A099448 A099449 * A099451 A099452 A099453

Keyword

easy,sign

Author

Paul Barry, Oct 16 2004

Status

approved