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A112575 Chebyshev transform of the second kind of the Pell numbers. 4

%I

%S 0,1,2,3,6,12,22,41,78,147,276,520,980,1845,3474,6543,12322,23204,

%T 43698,82293,154974,291847,549608,1035024,1949160,3670665,6912610,

%U 13017851,24515262,46167228,86942286,163730017,308336942,580661211,1093503228,2059289112

%N Chebyshev transform of the second kind of the Pell numbers.

%C 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)).

%H G. C. Greubel, <a href="/A112575/b112575.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-1)

%F G.f.: x/(1-2*x+x^2-2*x^3+x^4).

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k, k)*A000129(n-2k).

%F a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*C((n+k)/2, k)*(1+(-1)^(n-k))*A000129(k)/2.

%t 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 *)

%o (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

%o (Magma)

%o C<I>:= ComplexField();

%o [(&+[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

%Y Cf. A000129.

%K easy,nonn,changed

%O 0,3

%A _Paul Barry_, Sep 14 2005

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Last modified January 18 14:40 EST 2022. Contains 350455 sequences. (Running on oeis4.)