OFFSET
0,2
COMMENTS
Expansion level n is the first 2^n segments of the curve, and can be taken as a graph with visited points as vertices and segments as edges.
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).
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).
EXAMPLE
Curve n=4:
*--* *--*
| | | | Start S to end E along segments.
*--*--* *--* Distance a(4) = 12,
| | which is also graph diameter.
E *--* S--*
| |
*--*
PROG
(PARI) a(n) = if(n==0, 1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kevin Ryde, May 05 2021
STATUS
approved