OFFSET
0,2
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
G. C. Greubel, Table of n, a(n) for n = 0..1000
Dror Bar-Natan, 9 48, The Knot Atlas.
Index entries for linear recurrences with constant coefficients, signature (7,-11,7,-1).
FORMULA
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
MATHEMATICA
LinearRecurrence[{7, -11, 7, -1}, {1, 7, 38, 196, 1001}, 40] (* Harvey P. Dale, Jun 18 2021 *)
PROG
(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
(Sage)
def A099461_list(prec):
P.<x> = 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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 16 2004
STATUS
approved