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A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2). 15
1, 8, 56, 384, 2624, 17920, 122368, 835584, 5705728, 38961152, 266043392, 1816657920, 12404916224, 84706066432, 578409201664, 3949625081856, 26969727041536, 184160815677440, 1257528709087232, 8586943147278336 (list; graph; refs; listen; history; text; internal format)
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)

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]

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]

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 xrange(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. A084130, A251732, A251733.

Sequence in context: A272763 A199939 A003494 * A101596 A092521 A156088

Adjacent sequences:  A057081 A057082 A057083 * A057085 A057086 A057087

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 11 2000

STATUS

approved

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Last modified February 24 11:12 EST 2018. Contains 299603 sequences. (Running on oeis4.)