A235643 Total number of sides of a tetraflake-like fractal after n iterations, a(1) = 16 (see comments).

16, 68, 296, 1300, 5728, 25268, 111512, 492196, 2172592, 9590180, 42332936, 186866356, 824867584, 3641141012, 16072772984, 70948650820, 313182494032, 1382454408452, 6102448992488, 26937513095764, 118907935627168, 524885022092660, 2316954583165784

Offset: 1

Author

Kival Ngaokrajang, Apr 20 2014

Keywords

• nonn

Comments

Construction rule is same as for box and Vicsek fractals, but uses 6 boxes at initial stage (n = 1) and has only one symmetrical axis. The scale factor of these fractals is 1/3. The actual tetraflake fractals have scale factor of 1/2.

a(n) is the total number of sides at different lengths of a tetraflake-like fractal after n iterations. The perimeter (rounded down) is A235648(n). The total number of holes is A241271(n+1).

Links

• Kival Ngaokrajang, Illustration of initial terms
• Eric Weisstein's World of Mathematics, Box Fractal
• Wikipedia, n-flake
• Wikipedia, Vicsek Fractal
• Index entries for linear recurrences with constant coefficients, signature (6, -7).

Crossrefs

Cf. A241271, A235648.

Cf. A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).

Cf. A063628 (hexaflake).

Cf. A240916, A240917 (triflake-like); A238777 (tetraflake-like).

Programs

Mathematica

LinearRecurrence[{6,-7},{16,68},30] (* Harvey P. Dale, Jun 14 2014 *)

Formula

Conjecture from Colin Barker, Apr 21 2014: (Start)

a(n) = sqrt(2)*((3-sqrt(2))^n*(-1+sqrt(2))+(1+sqrt(2))*(3+sqrt(2))^n).

a(n) = 6*a(n-1)-7*a(n-2).

G.f.: 4*x*(-7*x+4) / (7*x^2-6*x+1). (End)

Extensions

More terms from Harvey P. Dale, Jun 14 2014