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 A178789 #62 Jun 15 2025 23:16:37
%S A178789 3,6,18,66,258,1026,4098,16386,65538,262146,1048578,4194306,16777218,
%T A178789 67108866,268435458,1073741826,4294967298,17179869186,68719476738,
%U A178789 274877906946,1099511627778,4398046511106,17592186044418,70368744177666
%N A178789 a(n) = 4^(n-1) + 2: Number of acute angles after n iterations of the Koch snowflake construction.
%C A178789 Starting from an equilateral triangle, at each step each straight segment is replaced by a "_/\_" shape of four segments of equal length, with the acute angle in the middle pointing to the exterior. The sequence counts the angles which are (i.e., already were) at both extremities, plus the one newly created acute angle in the middle of each former segment. At step n, there are 3*4^(n-1) straight segments, therefore a(n+1) = a(n) + 3*4^(n-1). - _M. F. Hasler_, Dec 17 2013
%H A178789 Vincenzo Librandi, <a href="/A178789/b178789.txt">Table of n, a(n) for n = 1..1000</a>
%H A178789 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A178789 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A178789 Mario Raso, <a href="https://iris.uniroma1.it/handle/11573/1732819">Integer Sequences in Cryptography: A New Generalized Family and its Application</a>, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 113.
%H A178789 Larry Riddle, <a href="http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm">Koch Curve</a>.
%H A178789 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F A178789 G.f.: 3*x*(1 - 3*x)/(1 - 5*x + 4*x^2).
%F A178789 a(n) = 3 * A047849(n-1).
%F A178789 a(n) = 2^(2*(n-1)) + 2. - _Vincenzo Librandi_, Feb 02 2013
%F A178789 a(n+1) = a(n) + 3*4^(n-1) = a(n) + A002001(n) for n > 0. - _M. F. Hasler_, Dec 17 2013
%F A178789 a(n) = 2 + A000302(n-1). - _Omar E. Pol_, Dec 18 2013
%p A178789 A178789:=n->2+4^(n-1); seq(A178789(n), n=1..30); # _Wesley Ivan Hurt_, Dec 17 2013
%t A178789 a=b=3;lst={a};Do[a=a+b;b*=4;AppendTo[lst,a],{n,40}];lst
%t A178789 Flatten[Table[2^(2*(n-1)) + 2, {n, 1, 50}]](* or *) CoefficientList[Series[(3 - 9*x)/(1 - 5*x + 4*x^2),{x, 0, 100}], x] (* _Vincenzo Librandi_, Feb 02 2013 *)
%o A178789 (Magma) [2^(2*(n-1)) + 2: n in [1..30]]; // _Vincenzo Librandi_, Feb 02 2013
%o A178789 (PARI) A178789=n->4^(n-1)+2 \\ - _M. F. Hasler_, Dec 17 2013
%Y A178789 Cf. A002001, A000302, A010502, A047849, A164346.
%K A178789 nonn,easy
%O A178789 1,1
%A A178789 _Vladimir Joseph Stephan Orlovsky_, Jun 14 2010