A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.
1, 2, 4, 8, 12, 18, 26, 36, 52, 70, 102, 136, 200, 266, 394, 524, 780, 1038, 1550, 2064, 3088, 4114, 6162, 8212, 12308, 16406, 24598, 32792, 49176, 65562, 98330, 131100, 196636, 262174, 393246, 524320, 786464, 1048610, 1572898, 2097188, 3145764, 4194342, 6291494
Offset: 0
Examples
Curve n=4:
*--* *--*
| | | | Start S to end E along segments.
*--*--* *--* Distance a(4) = 12,
| | which is also graph diameter.
E *--* S--*
| |
*--*
Links
- Kevin Ryde, Table of n, a(n) for n = 0..1000
- Kevin Ryde, Iterations of the Dragon Curve, see index "Diameter".
- Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-2,2).
Programs
-
PARI
a(n) = if(n==0,1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);
Formula
a(0) = 1.
a(2*n) = 3*2^n + 2*n - 4 = 2*A275970(n-1), for n>=1.
a(2*n+1) = 4*2^n + 2*n - 2 = 2*A083706(n).
a(n+1) - a(n) = 2*A228693(n), for n>=1.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6.
G.f.: (1 + x - x^2 + x^3 - 4*x^5) / ((1+x) * (1-x)^2 * (1-2*x^2)).
G.f.: 2 - (1/2)/(1+x) - (9/2)/(1-x) + 1/(1-x)^2 + (3 + 4*x)/(1 - 2*x^2).
Comments