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A251733 a(n) = 3^n*A077985(n-1), A077985(-1) = 0. Irrational parts of the integers in Q(sqrt(2)) giving the length of a Lévy C-curve variant at iteration step n. 6
0, 3, -18, 135, -972, 7047, -51030, 369603, -2676888, 19387755, -140418522, 1017000927, -7365772260, 53347641903, -386377801758, 2798395587675, -20267773741872, 146792202740307, -1063163180118690, 7700108905374903, -55769122053317628, 403915712468279895 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The rational parts are given in A251732.

Inspired by the Lévy C-curve, and generated using different construction rules as shown in the links.

The length of this variant Lévy 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) = A251732(n) = 3^n*A123335(n) and B(n) = a(n) = 3^n*A077985(n-1), with  A077985(-1) = 0. See the construction rule and the illustration in the links.

The total length of the Lévy C-curve after n iterations is sqrt(2^n), also an integer in Q(sqrt(2)). The fractal dimension of the Lévy C-curve is 2, but for this modified case it is log(3)/log(1+sqrt(2)) = 1.2464774357... .

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Kival Ngaokrajang, Construction rule, Illustration of modified Lévy C curve

Wikipedia, Lévy C curve

Index entries for linear recurrences with constant coefficients, signature (-6,9).

FORMULA

a(n) = 3^n*A077985(n-1), A077985(-1) = 0.

G.f.: 3*x /(1 + 6*x - 9*x^2). See the Colin Barker, Dec 07 2014 program.

a(n) = ((3*(-1+sqrt(2)))^n - (-3*(1+sqrt(2)))^n)/(2*sqrt(2)). - Colin Barker, Jan 21 2017

MATHEMATICA

LinearRecurrence[{-6, 9}, {0, 3}, 30] (* G. C. Greubel, Nov 18 2017 *)

PROG

(PARI) concat(0, Vec(-3*x / (9*x^2-6*x-1) + O(x^100))) \\ Colin Barker, Dec 07 2014

(MAGMA) [Round(((3*(-1+Sqrt(2)))^n - (-3*(1+Sqrt(2)))^n)/(2*Sqrt(2))): n in [0..30]]; // G. C. Greubel, Nov 18 2017

CROSSREFS

Cf. A123335, A077985, A251732, A017910.

Sequence in context: A074545 A192462 A168072 * A095776 A114178 A005159

Adjacent sequences:  A251730 A251731 A251732 * A251734 A251735 A251736

KEYWORD

sign,easy

AUTHOR

Kival Ngaokrajang, Dec 07 2014

EXTENSIONS

More terms from Colin Barker, Dec 07 2014

Edited: see A251732. - Wolfdieter Lang, Dec 07 2014

STATUS

approved

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Last modified November 15 18:29 EST 2019. Contains 329149 sequences. (Running on oeis4.)