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

1,1

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

Table of n, a(n) for n=1..23.

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).

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)

Mathematica

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

Prog

(Small Basic)
a[0] = 10
a[1] = 16
For n = 2 To 20
  t1 = a[n-1]*3
  t2 = a[n-2]*2
  t3 = 0
  If n >= 3 then
    for i = 3 to n
      t3 = t3 + a[i-3]*2*Math.Power(3, n-i+1)
    EndFor
  EndIf
  a[n] = t1 + t2 + t3
  TextWindow.Write(a[n-1]+", ")
EndFor

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).

Sequence in context: A178574 A005906 A247663 * A297886 A298491 A298291

Adjacent sequences: A235640 A235641 A235642 * A235644 A235645 A235646

Keyword

nonn

Author

Kival Ngaokrajang, Apr 20 2014

Extensions

More terms from Harvey P. Dale, Jun 14 2014

Status

approved