A178789 a(n) = 4^(n-1) + 2: Number of acute angles after n iterations of the Koch snowflake construction.

3, 6, 18, 66, 258, 1026, 4098, 16386, 65538, 262146, 1048578, 4194306, 16777218, 67108866, 268435458, 1073741826, 4294967298, 17179869186, 68719476738, 274877906946, 1099511627778, 4398046511106, 17592186044418, 70368744177666

Offset

1,1

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Larry Riddle, Koch Curve.

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Formula

G.f.: 3*x*(1 - 3*x)/(1 - 5*x + 4*x^2).

a(n) = 3 * A047849(n-1).

a(n) = 2^(2*(n-1)) + 2. - Vincenzo Librandi, Feb 02 2013

a(n+1) = a(n) + 3*4^(n-1) = a(n) + A002001(n) for n > 0. - M. F. Hasler, Dec 17 2013

a(n) = 2 + A000302(n-1). - Omar E. Pol, Dec 18 2013

Maple

A178789:=n->2+4^(n-1); seq(A178789(n), n=1..30); # Wesley Ivan Hurt, Dec 17 2013

Mathematica

a=b=3; lst={a}; Do[a=a+b; b*=4; AppendTo[lst, a], {n, 40}]; lst
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 *)

Prog

(Magma) [2^(2*(n-1)) + 2: n in [1..30]]; // Vincenzo Librandi, Feb 02 2013
(PARI) A178789=n->4^(n-1)+2  \\ - M. F. Hasler, Dec 17 2013

Crossrefs

Cf. A002001, A000302, A010502, A047849, A164346.

Sequence in context: A165200 A215635 A215634 * A102962 A076510 A220816

Adjacent sequences: A178786 A178787 A178788 * A178790 A178791 A178792

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Jun 14 2010

Status

approved