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.

A341030 Decimal expansion of the perimeter of the convex hull around the dragon curve fractal.

Original entry on oeis.org

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

Views

Author

Kevin Ryde, Feb 02 2021

Keywords

Comments

Benedek and Panzone determine the 10 vertices of the polygon which is the convex hull around the dragon fractal. The perimeter follows from these.

Examples

			4.1292731001...

Links

Crossrefs

Cf. A341029 (finite hull areas).

Programs

  • Mathematica
    RealDigits[2/3 + (1 + Sqrt[2])*(5 + Sqrt[13])/6, 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Formula

Equals 3/2 + (5/6)*sqrt(2) + (1/6)*sqrt(13) + (1/6)*sqrt(26).
Equals 2/3 + (1 + sqrt(2))*(5 + sqrt(13))/6.
Largest root of 81*x^4 - 486*x^3 + 693*x^2 - 282*x + 17 = 0 (all its roots are real).

A343485 Area of the convex hull around terdragon expansion level n, measured in unit triangles.

Original entry on oeis.org

0, 2, 8, 26, 86, 276, 856, 2586, 7826, 23628, 71128, 213546, 641246, 1925076, 5777416, 17333706, 52006586, 156031788, 468115048, 1404358266, 4213124006, 12639480276, 37918617976, 113755972026, 341268358946, 1023806051148, 3071419747768, 9214260306186
Offset: 0

Views

Author

Kevin Ryde, Apr 17 2021

Keywords

Comments

Expansion level n comprises the first 3^n segments of the curve.

Examples

			For n=1, the terdragon curve comprises 3 segments:
    @---@      Convex hull vertices are marked "@".
     \         They enclose an area of 2 unit triangles,
  @---@        so a(1) = 2.
.
For n=2, the terdragon curve comprises 9 segments:
    @---@
     \         Convex hull vertices are marked "@".
  @---*        They enclose an area of a(2) = 8
   \ / \       unit triangle equivalents.
    *---@
     \
  @---@

Links

Crossrefs

Cf. A343486 (fractal hull area), A341029 (dragon curve hull area).

Programs

  • PARI
    my(h=[30,46,22,50]); a(n) = if(n<2,2*n, (29*3^n - h[n%4+1]*3^(n\2))\24);

Formula

For n>=2, a(n) = (29/24)*3^n - (h/12)*3^floor(n/2) - (c/8) where h = 15,23,11,25 and c = 5,3,1,3 according as n == 0,1,2,3 (mod 4) respectively.
a(n) = 4*a(n-1) - 4*a(n-2) + 4*a(n-3) + 6*a(n-4) - 36*a(n-5) + 36*a(n-6) - 36*a(n-7) + 27*a(n-8), for n>=10.
G.f.: (2*x + 2*x^3 + 6*x^4 - 8*x^5 + 16*x^6 - 18*x^7 + 6*x^8 - 18*x^9) /( (1-x)*(1+x^2)*(1-9*x^4)*(1-3*x) ).
G.f.: (1/24)*( 16 + 16*x - 9/(1-x) - 6/(1+x^2) - (26+48*x)/(1-3*x^2) + (-4+2*x)/(1+3*x^2) + 29/(1-3*x) ).
Lim_{n->oo} a(n)/3^n = 29/24.

A362566 a(n) is the area of the smallest rectangle that the Harter-Heighway Dragon Curve will fit in after n doublings, starting with a segment of length 1.

Original entry on oeis.org

0, 1, 2, 6, 15, 42, 77, 180, 345, 806, 1457, 3276, 5985, 13462, 24257, 54060, 97665, 217686, 391937, 871596, 1570305, 3492182, 6286337, 13972140, 25155585, 55911766, 100642817, 223660716, 402612225, 894735702, 1610530817, 3578997420, 6442287105, 14316361046
Offset: 0

Views

Author

Nicolay Avilov, Apr 25 2023

Keywords

Comments

When constructing this sequence, the rectangles that are considered are those whose sides are parallel to the corresponding links of the dragon curve.

Examples

			See link:
a(3) = 2*3 = 6;
a(4) = 3*5 = 15;
a(5) = 6*7 = 42.

Links

Crossrefs

Cf. A014577, A341029.

Programs

  • Python
    x1, x2, y1, y2, ex, ey, a = 0, 1, 0, 0, 1, 0, [0]
for n in range(40):
    ex, ey = ex-ey, ey+ex
    x1r, x2r, y1r, y2r = y1+ex, y2+ex, -x2+ey, -x1+ey
    x1, x2, y1, y2 = min(x1, x1r), max(x2, x2r), min(y1, y1r), max(y2, y2r)
    a.append((x2-x1)*(y2-y1))
print(a) # Andrey Zabolotskiy, May 03 2023

Formula

From Andrey Zabolotskiy, Joerg Arndt and Kevin Ryde, May 03 2023: (Start)
G.f.: x * (1 + x + x^2 + 6*x^3 + 7*x^4 + 2*x^6) / ((1 - x) * (1 - 2*x) * (1 + 2*x) * (1 + x^2) * (1 - 2*x^2) * (1 + 2*x^2)).
a(n) =
(3*2^n - 5*2^(n/2) + 2) / 2 for n == 0 (mod 2),
(5*2^n - 9*2^((n-1)/2) + 2) / 3 for n == 1 (mod 4),
(5*2^n - 13*2^((n-1)/2) + 4) / 3 for n == 3 (mod 4). (End)

Extensions

Terms a(16) and beyond and a(0)=0 from Andrey Zabolotskiy, Apr 27 2023
Showing 1-3 of 3 results.

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