A251732 a(n) = 3^n*A123335(n). Rational parts of the integers in Q(sqrt(2)) giving the length of a Lévy C-curve variant at iteration step n.

1, -3, 27, -189, 1377, -9963, 72171, -522693, 3785697, -27418419, 198581787, -1438256493, 10416775041, -75444958683, 546420727467, -3957528992949, 28662960504897, -207595523965923, 1503539788339611, -10889598445730973, 78869448769442337, -571223078628232779

Offset: 0

Author

Kival Ngaokrajang, Dec 07 2014

Keywords

• sign
• easy

Comments

The irrational part is given in A251733.

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) = a(n) = 3^n*A123335(n) and B(n) = A251733(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)) (see a comment on A077957). 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... .

Examples

			The first lengths a(n) + A251733(n)*sqrt(2) are:
1, -3 + 3*sqrt(2), 27 - 18*sqrt(2), -189 + 135*sqrt(2), 1377 - 972*sqrt(2), -9963 + 7047*sqrt(2), 72171 - 51030*sqrt(2), -522693 + 369603*sqrt(2), 3785697 - 2676888*sqrt(2), -27418419 + 19387755*sqrt(2), 198581787 - 140418522*sqrt(2), ... - _Wolfdieter Lang_, Dec 08 2014
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• MathImages, Lévy's C-curve
• 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. A017910, A077985, A123335, A251733.

Programs

Magma

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

Mathematica

LinearRecurrence[{-6,9}, {1,-3}, 30] (* G. C. Greubel, Nov 18 2017 *)

PARI

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

Formula

a(n) = 3^n*A123335(n).

a(n) = -6*a(n-1) + 9*a(n-2). - Colin Barker, Dec 07 2014

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

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

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

Extensions

More terms from Colin Barker, Dec 07 2014

Edited: Name specified, Q(sqrt(2))remarks given earlier in a comment to a first version, MathImages link added. - Wolfdieter Lang, Dec 07 2014