login
A112575
Chebyshev transform of the second kind of the Pell numbers.
6
0, 1, 2, 3, 6, 12, 22, 41, 78, 147, 276, 520, 980, 1845, 3474, 6543, 12322, 23204, 43698, 82293, 154974, 291847, 549608, 1035024, 1949160, 3670665, 6912610, 13017851, 24515262, 46167228, 86942286, 163730017, 308336942, 580661211, 1093503228, 2059289112
OFFSET
0,3
COMMENTS
The Chebyshev transform of the second kind maps the sequence with g.f. g(x) to the sequence with g.f. (1/(1+x^2))g(x/(1+x^2)).
FORMULA
G.f.: x/(1-2*x+x^2-2*x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k, k)*A000129(n-2k).
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*C((n+k)/2, k)*(1+(-1)^(n-k))*A000129(k)/2.
MATHEMATICA
Table[Sum[(-1)^k*Binomial[n-k, k]*Fibonacci[n-2*k, 2], {k, 0, Floor[n/2]}], {n, 0, 40}] (* G. C. Greubel, Jan 14 2022 *)
PROG
(Sage) [sum((-1)^k*binomial(n-k, k)*lucas_number1(n-2*k, 2, -1) for k in (0..(n/2))) for n in (0..40)] # G. C. Greubel, Jan 14 2022
(Magma)
C<I>:= ComplexField();
[(&+[Binomial(n-k, k)*Round(I^(n-1)*Evaluate(ChebyshevU(n-2*k), -I)): k in [0..Floor(n/2)]]) : n in [0..40]]; // G. C. Greubel, Jan 14 2022
CROSSREFS
Cf. A000129.
Sequence in context: A047090 A185265 A018178 * A018079 A289920 A060985
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 14 2005
STATUS
approved