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A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2). 15

%I

%S 1,8,56,384,2624,17920,122368,835584,5705728,38961152,266043392,

%T 1816657920,12404916224,84706066432,578409201664,3949625081856,

%U 26969727041536,184160815677440,1257528709087232,8586943147278336

%N Scaled Chebyshev U-polynomials evaluated at sqrt(2).

%C From _Kival Ngaokrajang_, Dec 14 2014 (Start):

%C -2*a(n-1) is the irrational part of the integer in Q(sqrt 2) giving the length of a Levy C-curve variant L(n)=(2*(2- sqrt 2))^n at iteration step n. The length of this C-curve is an integer in the real quadratic number field Q(sqrt 2), namely L(n) = A(n)+B(n)*sqrt(2) with A(n) = A084130(n) and B(n) = -2*a(n-1). See the construction rule and the illustration in the links.

%C The fractal dimension of the Levy C-curve is 2, but for this modified case it is log(4)/log(2 + sqrt 2) = 1.1289527...

%C (End)

%D S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

%H G. C. Greubel, <a href="/A057084/b057084.txt">Table of n, a(n) for n = 0..1000</a>

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=-8.

%H W. Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=8.

%H Kival Ngaokrajang, <a href="/A057084/a057084.pdf">Illustration of construction rule and initial terms</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/L%C3%A9vy_C_curve">Lévy C curve</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8).

%F a(n) = 8*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.

%F a(n) = S(n, 2*sqrt(2))*(2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.

%F a(2*k) = A002315(k)*8^k, a(2*k+1) = A001109(k+1)*8^(k+1).

%F G.f.: 1/(1-8*x+8*x^2).

%F a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*8^k. [_Philippe Deléham_, Oct 28 2008]

%F a(n) = -(1/8)*[4-2*sqrt(2)]^(n+1)*sqrt(2)+(1/8)*sqrt(2)*[4+2*sqrt(2)]^(n+1), with n>=0. [_Paolo P. Lava_, Nov 20 2008]

%F Binomial transform of A002315. [Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009]

%e The first pairs [A(n),B(n)] determining the length L(n) are : [1, 0], [4, -2], [24, -16], [160, -112], [1088, -768], [7424, -5248], [50688, -35840], [346112, -244736], [2363392, -1671168], [16138240, -11411456], ... _Kival Ngaokrajang_, Dec 14 2014

%t Join[{a=1,b=8},Table[c=8*b-8*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 19 2011 *)

%t LinearRecurrence[{8,-8},{1,8},30] (* _Harvey P. Dale_, Feb 07 2015 *)

%o (Sage) [lucas_number1(n,8,8) for n in xrange(1, 21)] # _Zerinvary Lajos_, Apr 23 2009

%o (PARI) x='x+O('x^50); Vec(1/(1-8*x+8*x^2)) \\ _G. C. Greubel_, Jul 03 2017

%Y Cf. A084130, A251732, A251733.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 11 2000

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Last modified November 18 01:20 EST 2018. Contains 317279 sequences. (Running on oeis4.)