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.

0, 3, -18, 135, -972, 7047, -51030, 369603, -2676888, 19387755, -140418522, 1017000927, -7365772260, 53347641903, -386377801758, 2798395587675, -20267773741872, 146792202740307, -1063163180118690, 7700108905374903, -55769122053317628, 403915712468279895

Offset: 0

Author

Kival Ngaokrajang, Dec 07 2014

Keywords

• sign
• easy

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).

Crossrefs

Cf. A123335, A077985, A251732, A017910.

Programs

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

Mathematica

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

PARI

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

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

E.g.f.: exp(-3*x)*sinh(3*sqrt(2)*x)/sqrt(2). - Stefano Spezia, Feb 01 2023

Extensions

More terms from Colin Barker, Dec 07 2014

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