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
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
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
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