A240735 -id:A240735 - cp's OEIS frontend

A240733 a(n) = floor(6^n/(2+2*cos(Pi/9))^n).

1, 1, 2, 3, 5, 8, 13, 21, 32, 50, 78, 121, 187, 289, 448, 693, 1072, 1658, 2564, 3966, 6134, 9487, 14673, 22695, 35101, 54288, 83964, 129862, 200850, 310643, 480452, 743085, 1149282, 1777523, 2749182, 4251987, 6576279, 10171116, 15731022, 24330178, 37629950

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a nonaflake after n iterations, let a(0) = 1. The total number of sides is 9*A000400(n). The total number of holes is A002452(n). 2*cos(Pi/9) = 1.87938524... = diagonal b of nonagon (see comments in A123609).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of nonaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A002452, A123609, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240733:=n->floor(6^n/(2+2*cos(Pi/9))^n); seq(A240733(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(2 + 2*Cos[Pi/9])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(2+2*cos(Pi/9))^n)}
       for (n=0, 100, print1(a(n), ", "))

A240734 a(n) = floor(6^n/(2+sqrt(5))^n).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 11, 16, 22, 32, 46, 65, 92, 130, 185, 262, 371, 526, 745, 1056, 1496, 2119, 3001, 4251, 6021, 8528, 12080, 17110, 24236, 34328, 48622, 68869, 97547, 138166, 195700, 277191, 392616, 556104, 787670, 1115663, 1580234, 2238256, 3170284

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a decaflake after n iterations, let a(0) = 1. The total number of sides is 10*A000400(n). The total number of holes is A002275(n). 2 + sqrt(5) = A098317.

Kival Ngaokrajang, Illustration of decaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A002275, A098317, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240734:=n->floor(6^n/(2+sqrt(5))^n); seq(A240734(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(2 + Sqrt[5])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(2+sqrt(5))^n)}
       for (n=0, 100, print1(a(n), ", "))

A240840 Floor(6^n/(1+1/(2cos(5Pi/11)))^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 17, 22, 30, 40, 53, 71, 95, 126, 168, 223, 297, 395, 525, 698, 928, 1234, 1640, 2180, 2899, 3854, 5123, 6811, 9055, 12038, 16003, 21275, 28282, 37599, 49984, 66448, 88336, 117433, 156115

Offset: 0

Kival Ngaokrajang, Apr 13 2014

nonn

a(n) is the perimeter (rounded down) of a hendecaflake after n iterations, let a(0) = 1. The total number of sides is 11*A000400(n). The total number of holes is A016123(n), n >=1. 1/(2*cos(5*Pi/11)) = A231186.

Kival Ngaokrajang, Illustration of hendecaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A016123, A231186, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake), A240841 (tridecaflake).

A240840:=n->floor(6^n/(1+1/(2*cos(5*Pi/11)))^n); seq(A240840(n), n=0..50); # Wesley Ivan Hurt, Apr 13 2014

Table[Floor[6^n/(1 + 1/(2*Cos[5*Pi/11]))^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 13 2014 *)

{a(n)=floor(6^n/(1+1/(2*cos(5*Pi/11)))^n)}
       for (n=0, 100, print1(a(n), ", "))

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

Kival Ngaokrajang, Apr 13 2014

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.

Kival Ngaokrajang, Illustration of tridecaflake for n = 0..3
Wikipedia, n-flake

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

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

Table[Floor[8^n/(1 + 2*Sin[6*Pi/13]/(2*Sin[Pi/13]))^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 13 2014 *)

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

A240916 a(n) = 6a(n-1) + 22^(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

Kival Ngaokrajang, Apr 14 2014

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).

Kival Ngaokrajang, Illustration of triflake like fractal (Mitsubishi logo) for n = 0..3
Wikipedia, n-flake
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

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

Join[{0,0},LinearRecurrence[{9,-20,12},{3,24,158},30]] (* Harvey P. Dale, Jan 31 2015 *)

{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),", "))

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

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) = 23^(2n) - 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

Kival Ngaokrajang, Apr 14 2014

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.

Kival Ngaokrajang, Illustration of triflake like fractal (fan pattern) for n = 0..3
Wikipedia, n-flake,
Index entries for linear recurrences with constant coefficients, signature (13,-39, 27).

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

A240917:=n->2*3^(2*n) - 3*3^n + 1; seq(A240917(n), n=0..30); # Wesley Ivan Hurt, Apr 15 2014

Table[2*3^(2 n) - 3*3^n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 15 2014 *)

a(n)= 2*3^(2*n) - 3*3^n + 1
       for(n=0,100,print1(a(n),", "))

concat(0, Vec(-2*x*(3*x+5)/((x-1)*(3*x-1)*(9*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014

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

Kival Ngaokrajang, Apr 20 2014

nonn

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).

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).

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).

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

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)

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

Kival Ngaokrajang, Apr 20 2014

nonn

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).

Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Box Fractal
Wikipedia, n-flake
Wikipedia, Vicsek Fractal

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).

{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,", "));}

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

A240846 a(0)=0, a(1)=1, a(n) = a(n-1)*12 + 13.

Original entry on oeis.org

0, 1, 25, 313, 3769, 45241, 542905, 6514873, 78178489, 938141881, 11257702585, 135092431033, 1621109172409, 19453310068921, 233439720827065, 2801276649924793, 33615319799097529, 403383837589170361, 4840606051070044345, 58087272612840532153

Offset: 0

Kival Ngaokrajang, Apr 13 2014

a(n) is the total number of holes in a dodecaflake after n iterations.

G. C. Greubel, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A240735.

Concatenation([0], List([1..20], n-> (2^(2*n+1)*3^n - 13)/11 )); # G. C. Greubel, Feb 06 2020

[0] cat [(2^(2*n+1)*3^n - 13)/11: n in [1..20]]; // G. C. Greubel, Feb 06 2020

A240846:=n->`if`(n=0, 0, (2^(1+2*n)*3^n-13)/11); seq(A240846(n), n=0..20); # Wesley Ivan Hurt, Apr 13 2014

Join[{0},NestList[12#+13&,1,20]] (* or *) LinearRecurrence[{13,-12},{0,1,25},30] (* Harvey P. Dale, Sep 08 2017 *)

{a(n)=if(n<=0, 0, if(n<2, 1, a(n-1)*12+13))}
for(n=0, 20, print1(a(n),", "))

[0]+[(2^(2*n+1)*3^n - 13)/11 for n in (1..20)] # G. C. Greubel, Feb 06 2020

From Colin Barker, Apr 13 2014: (Start)
a(n) = (2^(2*n+1)*3^n - 13)/11 for n>0.
a(n) = 13*a(n-1) - 12*a(n-2) for n>2.
G.f.: x*(1+12*x) /((1-x)*(1-12*x)). (End).
E.g.f.: (11 - 13*exp(x) + 2*exp(12*x))/11. - G. C. Greubel, Feb 06 2020

Name changed by Colin Barker, Apr 13 2014

cp's OEIS Frontend

A240733 a(n) = floor(6^n/(2+2*cos(Pi/9))^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A240734 a(n) = floor(6^n/(2+sqrt(5))^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A240840 Floor(6^n/(1+1/(2*cos(5*Pi/11)))^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A240840 Floor(6^n/(1+1/(2cos(5Pi/11)))^n).

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

A240916 a(n) = 6a(n-1) + 22^(n-1) - 2 for n > 2, a(0) = a(1) = 0, a(2) = 3.

A240917 a(n) = 23^(2n) - 3*3^n + 1.