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 1-5 of 5 results.

A240841 a(n) = floor(8^n/(1+2*sin(6*Pi/13)/(2*sin(Pi/13)))^n).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 14, 21, 34, 52, 82, 127, 198, 308, 478, 744, 1156, 1796, 2792, 4339, 6742, 10477, 16282, 25302, 39318, 61100, 94947, 147545, 229281, 356295, 553672, 860388, 1337014, 2077676, 3228640, 5017200, 7796562, 12115600, 18827241, 29256909, 45464268
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 13 2014

Keywords

Comments

a(n) is the perimeter (rounded down) of a tridecaflake after n iterations, let a(0) = 1. The total number of sides is 13*A001018(n). The total number of holes is A091030(n), n >= 1.

Links

Crossrefs

Cf. A001018, A091030, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake).

Programs

  • Maple
    A240841:=n->floor(8^n/(1+2*sin(6*Pi/13)/(2*sin(Pi/13)))^n); seq(A240841(n), n=0..50); # Wesley Ivan Hurt, Apr 13 2014
  • Mathematica
    Table[Floor[8^n/(1 + 2*Sin[6*Pi/13]/(2*Sin[Pi/13]))^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 13 2014 *)
  • PARI
    {a(n)=floor(8^n/(1+2*sin(6*Pi/13)/(2*sin(Pi/13)))^n)}

A240916 a(n) = 6*a(n-1) + 2*2^(n-1) - 2 for n > 2, a(0) = a(1) = 0, a(2) = 3.

Original entry on oeis.org

0, 0, 3, 24, 158, 978, 5930, 35706, 214490, 1287450, 7725722, 46356378, 278142362, 1668862362, 10013190554, 60079176090, 360475122074, 2162850863514, 12977105443226, 77862633183642, 467175800150426, 2803054802999706, 16818328822192538
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 14 2014

Keywords

Comments

a(n) is the total number of holes of a triflake-like fractal (Mitsubishi logo) after n iterations. The scale factor for this case is 1/3, but for the actual triflake case, it is 1/2, i.e., Sierpiński triangle. The total number of sides is 3*A000302(n). The perimeter (rounded down) is A064628(n).

Links

Crossrefs

Cf. A000302, A064628, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).

Programs

  • Mathematica
    Join[{0,0},LinearRecurrence[{9,-20,12},{3,24,158},30]] (* Harvey P. Dale, Jan 31 2015 *)
  • PARI
    {a(n)=if(n<=0, 0, if(n<2, 0, if(n<3, 3, 6*a(n-1)+2*2^(n-1)-2)))}
  for(n=0,100,print1(a(n),", "))
  • PARI
    concat([0,0], Vec(-x^2*(2*x^2-3*x+3)/((x-1)*(2*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014

Formula

From Colin Barker, Apr 15 2014: (Start)
a(n) = (72-45*2^(1+n)+23*6^n)/180 for n>1.
a(n) = 9*a(n-1)-20*a(n-2)+12*a(n-3) for n>4.
G.f.: -x^2*(2*x^2-3*x+3) / ((x-1)*(2*x-1)*(6*x-1)). (End).

A240917 a(n) = 2*3^(2*n) - 3*3^n + 1.

Original entry on oeis.org

0, 10, 136, 1378, 12880, 117370, 1060696, 9559378, 86073760, 774781930, 6973391656, 62761587778, 564857478640, 5083726873690, 45753570561016, 411782221142578, 3706040248563520, 33354363011912650, 300189269431736776, 2701703431859199778
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 14 2014

Keywords

Comments

a(n) is the total number of holes of a triflake-like fractal (fan pattern) after n iterations. The scale factor for this case is 1/3, but for the actual triflake case, it is 1/2, i.e., Sierpiński triangle. The total number of sides is 3*(A198643-1). The perimeter seems to converge to 10/6.

Links

Crossrefs

Cf. A198643, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).

Programs

  • Maple
    A240917:=n->2*3^(2*n) - 3*3^n + 1; seq(A240917(n), n=0..30); # Wesley Ivan Hurt, Apr 15 2014
  • Mathematica
    Table[2*3^(2 n) - 3*3^n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 15 2014 *)
  • PARI
    a(n)= 2*3^(2*n) - 3*3^n + 1
       for(n=0,100,print1(a(n),", "))
  • PARI
    concat(0, Vec(-2*x*(3*x+5)/((x-1)*(3*x-1)*(9*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014

Formula

a(n) = 2*A007742(A003462(n)).
a(n) = 9*(a(n-1) + 2*A048473(n-1)) + 1.
From Colin Barker, Apr 15 2014: (Start)
a(n) = 1-3^(1+n)+2*9^n.
a(n) = 13*a(n-1)-39*a(n-2)+27*a(n-3).
G.f.: -2*x*(3*x+5) / ((x-1)*(3*x-1)*(9*x-1)). (End).

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

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 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

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 1-5 of 5 results.

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