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 1-4 of 4 results.

A349008 Area of the convex hull around R5 dragon curve expansion level n.

Original entry on oeis.org

0, 2, 16, 106, 578, 2954, 15064, 75908, 380334, 1904330, 9528618, 47654840, 238295096, 1191556256, 5957936770, 29789860126, 148950201902, 744752899780, 3723767329212, 18618843605284, 93094240114350, 465471240742354, 2327356261817746, 11636781564585616
Offset: 0

Views

Author

Kevin Ryde, Nov 06 2021

Keywords

Comments

Expansion level n is the first 5^n segments of the curve and so the hull is around X,Y points located at X=A349195(t), Y=A349196(t) for t = 0..5^n.

Examples

			For n=2 the curve is:
  @--@
  |
  *--*  *--*  *--@      Hull vertices "@".
     |  |  |  |  |      Hull area a(2) = 16.
  *--*--*--*--*--*
  |  |  |  |  |
  @--*  *--*  *--*
                 |
              @--@

Links

Crossrefs

Cf. A349195, A349196 (coordinates).
Cf. A349009 (fractal area).

Programs

  • PARI
    \\ See links.

Formula

For n>=2, a(n) = 17*5^(n-2) - 1 + Sum_{j=1..n-2} ( (3*5^(n-2-j)-1)*HAgrow(2*b^j) + 2*5^(n-2-j)*HAgrow((4-i)*b^j) ),
where complex b=1+2*i and
HAgrow(z) = MinReIm(ShearRe(RotQ(z))),
MinReIm(z) = min(abs(Re z),abs(Im z)),
ShearRe(z) = z + Re(z),
RotQ(z) = z if sign(Re z) = sign(Im z) or RotQ(z) = z*i otherwise.

A349010 Decimal expansion of the perimeter of the convex hull around the R5 dragon fractal.

Original entry on oeis.org

3, 7, 4, 3, 6, 6, 9, 4, 4, 1, 2, 4, 6, 9, 8, 0, 0, 9, 8, 4, 9, 2, 2, 3, 3, 4, 0, 9, 8, 8, 2, 1, 4, 1, 3, 0, 4, 2, 3, 5, 1, 2, 7, 0, 3, 3, 9, 9, 4, 0, 5, 8, 4, 6, 3, 4, 6, 7, 8, 1, 2, 3, 2, 7, 4, 0, 2, 1, 9, 0, 1, 0, 8, 7, 9, 0, 1, 7, 0, 5, 9, 7, 2, 0, 0, 9, 1, 1, 2, 2, 3, 6, 7, 5, 7, 8, 6, 6, 2, 8, 6, 6, 1, 6, 2
Offset: 1

Views

Author

Kevin Ryde, Nov 06 2021

Keywords

Comments

The fractal is taken scaled to unit length from curve start to end.
With complex b = 1+2i, the hull sides are a countably infinite set: +-(4-i)/b^2, +-2/b^2, and 2*i^d/b^k for d=0..3 and k>=3. The sum of their magnitudes is the present constant.

Examples

			3.7436694412469800984922334098821413...

Links

Crossrefs

Cf. A349009 (area).

Programs

  • Mathematica
    RealDigits[(6 + 2*Sqrt[5] + 2*Sqrt[17])/5, 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)
  • PARI
    my(c=352+32*quadgen(5*17*4)); a_vector(len) = my(s=10^(len-2)); digits(sqrtint(floor(c*s^2)) + floor(12*s));

Formula

Equals (6 + 2*sqrt(5) + 2*sqrt(17)) / 5.
Equals (sqrt(8*sqrt(5*17) + 88) + 6) / 5.
Largest root of 625*x^4 - 3000*x^3 + 1000*x^2 + 6240*x - 2736 = 0 (all roots are real).

A349195 a(n) is the X-coordinate of the n-th point of the R5 dragon curve; A349196 gives Y-coordinates.

Original entry on oeis.org

0, 1, 1, 0, 0, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, -2, -2, -3, -3, -4, -4, -3, -3, -4, -4, -3, -3, -4, -4, -5, -5, -6, -6, -5, -5, -6, -6, -5, -5, -4, -4, -5, -5, -4, -4, -5, -5, -6, -6, -7, -7, -8, -8, -7, -7, -8, -8, -7, -7, -6, -6, -5, -5, -6, -6, -5, -5
Offset: 0

Views

Author

Rémy Sigrist, Nov 10 2021

Keywords

Comments

The R5 dragon curve can be represented using an L-system.

Examples

			The R5 dragon curve starts as follows:
         +-----+
       24|   25
         |
         |
         +-----+     +-----+     +-----+
       23    22|   11|   10|    7|    6|
               |     |     |     |     |
             21|   12|    9|    8|     |
         +-----+-----+-----+-----+-----+
       20|   17|   16|   13|    4|    5
         |     |     |     |     |
         |     |     |     |     |
         +-----+     +-----+     +-----+
       19    18    15    14     3     2|
                                       |
                                       |
                                 +-----+
                                0     1
- so a(0) = a(3) = a(4) = a(7) = a(8) = 0,
     a(1) = a(2) = a(5) = a(6) = 1,
     a(9) = a(10) = a(13) = a(14) = -1.

Links

Crossrefs

Cf. A006495, A349008, A349009, A349010, A349196.

Programs

  • PARI
    See Links section.

Formula

a(5^k) = A006495(k) for any k >= 0.

A349196 a(n) is the Y-coordinate of the n-th point of the R5 dragon curve; A349195 gives X-coordinates.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 4, 4, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, -1, -1, 0, 0, -1, -1, -2, -2, -3, -3, -2, -2, -3, -3
Offset: 0

Views

Author

Rémy Sigrist, Nov 10 2021

Keywords

Comments

The R5 dragon curve can be represented using an L-system.

Examples

			The R5 dragon curve starts as follows:
         +-----+
       24|   25
         |
         |
         +-----+     +-----+     +-----+
       23    22|   11|   10|    7|    6|
               |     |     |     |     |
             21|   12|    9|    8|     |
         +-----+-----+-----+-----+-----+
       20|   17|   16|   13|    4|    5
         |     |     |     |     |
         |     |     |     |     |
         +-----+     +-----+     +-----+
       19    18    15    14     3     2|
                                       |
                                       |
                                 +-----+
                                0     1
- so a(0) = a(1) = 0,
     a(2) = a(3) = a(14) = a(15) = a(18) = a(19) = 1.

Links

Crossrefs

Cf. A006496, A349008, A349009, A349010, A349195.

Programs

  • PARI
    See Links section.

Formula

a(5^k) = A006496(k) for any k >= 0.
Showing 1-4 of 4 results.

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