This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A251732 #35 Feb 01 2023 14:29:24
%S A251732 1,-3,27,-189,1377,-9963,72171,-522693,3785697,-27418419,198581787,
%T A251732 -1438256493,10416775041,-75444958683,546420727467,-3957528992949,
%U A251732 28662960504897,-207595523965923,1503539788339611,-10889598445730973,78869448769442337,-571223078628232779
%N 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.
%C A251732 The irrational part is given in A251733.
%C A251732 Inspired by the Lévy C-curve, and generated using different construction rules as shown in the links.
%C A251732 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.
%C A251732 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... .
%H A251732 Colin Barker, <a href="/A251732/b251732.txt">Table of n, a(n) for n = 0..1000</a>
%H A251732 MathImages, <a href="http://mathforum.org/mathimages/index.php/Lévy's_C-curve">Lévy's C-curve</a>
%H A251732 Kival Ngaokrajang, <a href="/A251732/a251732.pdf">Construction rule</a>, <a href="/A251732/a251732_1.pdf">Illustration of modified Lévy C curve</a>
%H A251732 Wikipedia, <a href="http://en.wikipedia.org/wiki/L%C3%A9vy_C_curve">Lévy C curve</a>
%H A251732 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-6,9).
%F A251732 a(n) = 3^n*A123335(n).
%F A251732 a(n) = -6*a(n-1) + 9*a(n-2). - _Colin Barker_, Dec 07 2014
%F A251732 G.f.: -(3*x+1)/(9*x^2-6*x-1). - _Colin Barker_, Dec 07 2014
%F A251732 a(n) = ((3*(-1+sqrt(2)))^n + (-3*(1+sqrt(2)))^n) / 2. - _Colin Barker_, Jan 21 2017
%F A251732 E.g.f.: exp(-3*x)*cosh(3*sqrt(2)*x). - _Stefano Spezia_, Feb 01 2023
%e A251732 The first lengths a(n) + A251733(n)*sqrt(2) are:
%e A251732 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
%t A251732 LinearRecurrence[{-6,9}, {1,-3}, 30] (* _G. C. Greubel_, Nov 18 2017 *)
%o A251732 (PARI) Vec(-(3*x+1) / (9*x^2-6*x-1) + O(x^100)) \\ _Colin Barker_, Dec 07 2014
%o A251732 (Magma) [Round(((3*(-1+Sqrt(2)))^n + (-3*(1+Sqrt(2)))^n)/2): n in [0..30]]; // _G. C. Greubel_, Nov 18 2017
%Y A251732 Cf. A017910, A077985, A123335, A251733.
%K A251732 sign,easy
%O A251732 0,2
%A A251732 _Kival Ngaokrajang_, Dec 07 2014
%E A251732 More terms from _Colin Barker_, Dec 07 2014
%E A251732 Edited: Name specified, Q(sqrt(2))remarks given earlier in a comment to a first version, MathImages link added. - _Wolfdieter Lang_, Dec 07 2014