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

0,2

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

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).

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]

Example

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

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 *)

Prog

(Sage) [lucas_number1(n, 8, 8) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009
(PARI) x='x+O('x^50); Vec(1/(1-8*x+8*x^2)) \\ G. C. Greubel, Jul 03 2017

Crossrefs

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

Sequence in context: A272763 A199939 A003494 * A101596 A327834 A092521

Adjacent sequences: A057081 A057082 A057083 * A057085 A057086 A057087

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 11 2000

Status

approved