cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

Kival Ngaokrajang, Apr 20 2014

Keywords

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

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

A241271 a(n) = 6*a(n-1) + 3*(2^(n-2)-1) for n > 2, a(0)=a(1)=a(2)=0.

Original entry on oeis.org

0, 0, 0, 3, 27, 183, 1143, 6951, 41895, 251751, 1511271, 9069159, 54418023, 326514279, 1959097959, 11754612327, 70527723111, 423166436967, 2538998818407, 15233993303655, 91403960608359, 548423765223015, 3290542594483815, 19743255573194343, 118459533451748967
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 18 2014

Keywords

Comments

a(n) is the total number of irregular polygon holes of a triflake-like fractal (A240916) after n iterations. A240916(n) - a(n) is the total number of rhombic holes.

Links

Crossrefs

Cf. A240916.

Programs

  • Maple
    A241271:=n->`if`(n=0, 0, (24-15*2^n+6^n)/40); seq(A241271(n), n=0..40); # Wesley Ivan Hurt, Apr 19 2014
  • Mathematica
    CoefficientList[Series[-3 x^3/((x - 1) (2 x - 1) (6 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)
LinearRecurrence[{9,-20,12},{0,0,0,3},30] (* Harvey P. Dale, Dec 28 2021 *)
  • PARI
    a(n) = if(n<=0,0,if(n<2,0,if(n<3,0,a(n-1)*6+3*(2^(n-2)-1))))
       for(n=0,100,print1(a(n),", "))
  • PARI
    concat([0,0,0], Vec(-3*x^3/((x-1)*(2*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Apr 18 2014

Formula

a(n) = (24-15*2^n+6^n)/40 for n>0. G.f.: -3*x^3 / ((x-1)*(2*x-1)*(6*x-1)). - Colin Barker, Apr 18 2014

A235648 Perimeter (rounded down) of a tetraflake-like fractal after n iterations, a(1) = 1 (see comments).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 16, 25, 39, 61, 97, 155, 249, 404, 657, 1073, 1759, 2892, 4768, 7877, 13036, 21602, 35838, 59508, 98885, 164416, 273502, 455137, 757628, 1261470, 2100791, 3499106, 5828894, 9710891, 16179575, 26958966, 44922289, 74858052, 124746848, 207889317
Offset: 1

Views

Author

Kival Ngaokrajang, Apr 20 2014

Keywords

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 a scale factor of 1/2.
The total number of sides at different lengths of a tetraflake-like fractal after n iterations is A235643(n). The total number of holes is A241271(n+1).

Links

Crossrefs

Cf. A241271, A235643.
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

  • PARI
    {a=18;c=1;print1(1,", "); for (n=1,50, c=4*c+3^(n-1); a=5*a-2*c; aa=floor((a*(1/3)^n)/18); print1(aa,", "));}

Formula

Floor((5*a(n-1)-2*(4*c(n-1)+3^(n-1)))/18) for n >1, a(1)=18, c(1)=1.
Showing 1-3 of 3 results.

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