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A099459
Expansion of 1/(1 - 7*x + 9*x^2).
5
1, 7, 40, 217, 1159, 6160, 32689, 173383, 919480, 4875913, 25856071, 137109280, 727060321, 3855438727, 20444528200, 108412748857, 574888488199, 3048504677680, 16165536349969, 85722212350663, 454565659304920
OFFSET
0,2
COMMENTS
Associated to the knot 9_48 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099460 and A099461.
LINKS
Dror Bar-Natan, 9 48, The Knot Atlas.
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
FORMULA
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-9)^k*7^(n-2*k).
a(n) = Sum{k=0..n} binomial(2*n-k+1, k) * 3^k. - Paul Barry, Jan 17 2005
a(n) = 7*a(n-1) - 9*a(n-2), n >= 2. - Vincenzo Librandi, Mar 18 2011
a(n) = ((7 + sqrt(13))^(n+1) - (7 - sqrt(13))^(n+1))/(2^(n+1)*sqrt(13)). - Rolf Pleisch, May 19 2011
a(n) = 3^(n-1)*ChebyshevU(n-1, 7/6). - G. C. Greubel, Nov 18 2021
MATHEMATICA
LinearRecurrence[{7, -9}, {1, 7}, 30] (* Harvey P. Dale, Jan 06 2012 *)
PROG
(Sage) [lucas_number1(n, 7, 9) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009
(Magma) [n le 2 select 7^(n-1) else 7*Self(n-1) -9*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 18 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 16 2004
STATUS
approved