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

1, 8, 56, 384, 2624, 17920, 122368, 835584, 5705728, 38961152, 266043392, 1816657920, 12404916224, 84706066432, 578409201664, 3949625081856, 26969727041536, 184160815677440, 1257528709087232, 8586943147278336

Offset: 0

Author

Wolfdieter Lang, Aug 11 2000

Keywords

• nonn
• easy

Comments

From Kival Ngaokrajang, Dec 14 2014 (Start):

-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.

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...

(End)

For lim_{n->oo} a(n+1)/a(n) = 2*(2 + sqrt(2)) = 6.82842... see A365823. - Wolfdieter Lang, Nov 15 2023

Examples

			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
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=-8.
• W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=8.
• Kival Ngaokrajang, Illustration of construction rule and initial terms
• Wikipedia, Lévy C curve
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (8,-8).

Crossrefs

Cf. A001109, A002315, A049310, A084130, A109466, A251732, A251733, A365823.

Programs

Mathematica

Join[{a=1,b=8},Table[c=8*b-8*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-8},{1,8},30] (* Harvey P. Dale, Feb 07 2015 *)

PARI

x='x+O('x^50); Vec(1/(1-8*x+8*x^2)) \\ G. C. Greubel, Jul 03 2017

Sage

[lucas_number1(n,8,8) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

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

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.

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

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

a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*8^k. [Philippe Deléham, Oct 28 2008]

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