A343485 Area of the convex hull around terdragon expansion level n, measured in unit triangles.
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
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
- Kevin Ryde, Table of n, a(n) for n = 0..500
- Kevin Ryde, Iterations of the Terdragon Curve, see index "HA".
- Index entries for linear recurrences with constant coefficients, signature (4,-4,4,6,-36,36,-36,27).
Programs
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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.
Comments